Two of my favorite mechanical engineering instructors worked for G&E and Pratt-Whitney designing jet engines. They said a common factor of safety for airplanes was 1.05; in other words, just barely above the theoretical material strength. So, although 3% may not seem like a lot when you’re talking about, say, an American muscle car, but it’s quite a bit to an aircraft designer.
Well, there’s a difference between what the aircraft is capable of doing, and what is legal. Legally, certification-wise, the maximum weights allowed for an aircraft are always some percentage below what the structure is theoretically capable of lifting. Redundancy and Fudge factor in the direction of safety is a constant in aviation.
Would those extra 6 people cause the aircraft to crash? On an aircraft the size of an A310, that’s highly unlikely. Did those extra 6 people exceed what the legal limits of the published limitations for that aircraft? Probably- and if it’s not legal, something has to go, either people or bags, and it’s often less costly for an airline to promise you a seat on a later flight and some sort of ‘company money’ (like Delta Dollars), than it is for them to pay for a baggage delivery service to track you and several other passengers down to deliver your offloaded baggage.
No, there’s not likely to be such a low tolerance for a commercial flight. Keep in mind that the physical maximum take-off weight may be higher than the maximum take-off weight imposed by company policy or legal operating thresholds. It’s possible your flight could handle additional weight just fine, but if the standards of the company say that the max is x lbs, then that’s the cut-off.
ETA: Also, what Av8trix said.
Bah. repost. Disregard.
My response would be the same, except for the second sentence. I just wouldn’t want everyone to know how much I weigh! 
I also am fairly certain that with most aircraft, the weight (aircraft and luggage weight are known, passenger weight is estimated) and centre-of-gravity are calculated using a formula sheet, such as this one.
The debate as to whether actual weights or average weights should be used is a pretty big one in aircraft safety, because it really can play a huge role in accidents. The traveling public argues strongly against actual weights being used, but they really would be safer. I believe the “average passenger weight” that is used in calculations now is quite a bit higher than 170lbs… I think it’s closer to 200 for men and 180 for women or something. A too-low estimate, and an aft-centre of gravity, were factors inthis crash and a possible factor in this one (aft-CG anyways) (WARNING: PDF). The problem is that it is very hard to get an accurate “average” weight for passengers. Either they take fewer passengers on/ask people to change flights, or they leave your luggage behind (which the traveling public also argues strongly about… there is no way to win this argument from a safety perspective!)
The aircraft are designed with fairly tight tolerances for things like weight; I know a small, +5lb change to a part on one aircraft took upwards of two years of calculations, tests, discussions with airworthiness authorities and various levels of documentation and paperwork and bureaucracy to get approved. The same aircraft type, at present, is not certified to carry a full fuel and full passenger load… very few aircraft types are. Does this mean the plane physically can’t do it? No. In fact it very likely can. But you can’t do it legally, and it’s going to take a buttload of work for the manufacturer to prove that you can! In the meantime, better safe than sorry, because while people may bitch about being kicked off a plane or having their luggage left behind, people bitch a hell of a lot more when a plan crashes!
Regarding the A310: there are a couple versions of it, with different numbers of passengers depending on the class configuration selected by the airline. If the original plane was intended to be an A310-300, and it had to be replaced by an A310-100, it might have had the same physical number of seats, but the former has an MTOW of 361,600 lb while the latter is limited to 312,342 lb. Put in the same amount of fuel for the same flight plan, and you can see where you might be running up against problems when it comes to passenger weight.
Sorry this post is a little bit rambly… I tend to do that 
Honestly, you think people don’t have an approximate idea? Of course, regardless of what you say, the pilot/boarding attendant’s going to have an idea, so if they think you’re lying, everyone doesn’t have to die, but they appreciate an honest answer, too. And if you’re big enough to worry about it, when the attendant asks, surprise them and state it proud and loud. At least then if there is a problem during the flight, nobody will be blaming “that lying @#$!%!!!”. 
That said, larger planes tend to even themselves out, it’s unlikely you’re going to have all your heavyweights sitting starboard aft. The smaller planes are more of an issue, but where you sit rarely makes a difference on smaller commercial planes. On the other hand, a lot of those sub-20 passenger planes have multiple luggage compartments. So if the plane’s risking a balance problem, baggage handlers will simply put more weight in a different compartment to balance the load.
Even so, the issue isn’t so much takeoff, at least not on any common carrier. Most planes, as has been stated, are well rated for commercial passenger traffic. Remember, while most are designed for commercial passengers, many are sold as cargo carriers as well. So while aunt Edna may tip the scales at home, she’s probably no comparison to most cargo that would take up the same size and space on an identical freight plane (including seat, leg room, kitchen and bathroom, and other amenities airlines provide for their human passengers).
A more serious issue is landing. Not so much the plane handling the weight, but the pilot’s adjusting for varying weight loads and balance issues. On a larger plane, things tend to work themselves out, and there’s more room for error. On smaller planes you might have a less experienced pilot, smaller, more difficult runway, and if the destination is a smaller airport, a less experienced or nonexistent ATC to offer the pilot a heads up about surface conditions.
If the plane’s heavy forward or aft, or side to side, that can make a difference in how the pilot makes his final approach. Most of the time, it’s already planned for, but if conditions suddenly change, it helps if the pilot knows what he’s dealing with. Remember, like walking, landing is basically a controlled fall. Unlike walking, when you screw up, it hurts more.
And for real puddle jumpers (if you can talk to the pilot while s/he’s flying, that’s you), the weight issue of the handful of passengers might be the largest variable the pilot has to compensate for. And heck, it gives you something do while your passengers are busy trying to figure out where seat 6 is on a 8 passenger plane.
The answer to the fundamental question asked by the OP is that yes aircraft can physically fit more weight on board than they can legally carry. This is most likely done to allow flexibility in how the airlines distribute the payload.
To rephrase some of what’s already been posted: The max take-off weight of an aircraft is determined by a number of factors. The max weight ultimately must be the lessor of the maximum structural weight (the max weight the aircraft can physically handle), the maximum weight that allows a take-off from the specific runway, the maximum weight that allows for adequate climb performance after take-off, and finally the max weight that allows the aircraft to be below its maximum landing weight on arrival at its destination.
When the max weight is found for the day, the calculated weight of the aircraft must not be more than the maximum. This may mean that on an aircraft with a max weight of 200,000 pounds, if the calculated weight is 200,220 pounds, you are going to have to drop some bags or a couple of people off.
For large aircraft this calculation is fairly complex as it must take into account performance with one engine failed and the ability to climb above obstacles in the vicinity of the aerodrome in that configuration. To avoid having pilots spend hours working this stuff out before each flight the operator will have a performance division who produce take-off and landing charts specific to each runway at each airfield. The pilot uses the current temperature and wind conditions and can find the maximum allowable take-off weight. The chart also provides critical speeds for take-off, one of which being the point at which you can still abort (“V1”.) Note that they do not think in terms of “if I am not flying by point X, I need to abort.” Large aircraft have guaranteed performance. Provided it is correctly configured, the engines are producing adequate thrust, and they are not overweight, the aircraft will take-off. Also, by the time they should be flying, there is no longer enough runway to abort.
I have boarded a small plane, in South America, where they were weighing the passengers at boarding. It was definitely disconcerting, but I did get on the plane. The mister was too busy deciding who he’d eat first if the plane crashed:smack:!
I was also on a small plane, once in Indonesia, where our luggage had all been piled in the cabin, somewhat loosely stacked in the first 4-6 seats. The same plane had a problem with the cockpit door. It was one of those little ass end loading planes, the hand rail being, well, balers twine actually. As we’re barreling down the runway, the luggage is starting to shift about, and the cockpit door keeps flying open and you can see the pilot pulling with all his might, on the controls, the end of the runway clearly visible through the windshield. It upset me so much I couldn’t enjoy my boxed lunch, though we did arrive safely. My understanding was that, chances were, they had the cargo bin full of some sort of expensive air frieght delivery.