Yes, I realize the answer depends on both the part and the car. Suppose both are relatively common, like a belt for a Model A.
I guess the underlying question is, owning antique car mean spending lots of time searching for parts?
Hemmings Motor News is recommended highly by Tom and Ray.
I own a 1966 Dodge Dart. Not quite an antique car, but old enough that some parts are tough to find.
It depends on how old the car is. Something as old as a Model A may be next to impossible to find parts for at normal auto parts stores, although V-belts have changed so little that if you knew the length you needed you probably could get one at AutoZone. For cars from the '60s or later, you can often find most of the parts to keep it running at chain parts stores, although they may not necessarily be the correct shape, color, or have the right numbers stamped on them to please collectors.
If you’re trying to keep a Model A on the road, it’s doable, but with a bit more effort at finding parts (and a little less effort at installing them, in many cases). Often it is easier to repair a part if the basic components are still intact than to try and track down a duplicate. There are places out there that will sell or rebuild most of the mechanical parts; you’ll want to check Hemmings Motor News and the various enthusiast websites for them. The really tough things to find are trim items - often these changed from year to year, and nobody makes new ones.
If you start with one in good shape and don’t try to use it as your commuter car, you won’t have that much time spent looking for parts. But if you start with the stereotypical basket case found in a farmer’s field, you will have to spend countless hours tracking down or fabricating all the little missing parts to get it back together, unless you’re building a hot rod with a modern drivetrain and suspension.
Like you said, it does depend a lot on the part and the car. Our family has a 1966 Chevelle SS that we restored, and we had a heck of a time finding a decent hood for it. Parts that are interchangeable between models are pretty easy to find, such as a ball joint for my car that fits almost every A-body between 1964 and 1972. Parts that are unique to a certain model, like my hood, can be a real pain.
You will be able to find most stuff easily if your car is a reasonably desirable model, but there will always be those 1 or 2 parts you will have to comb the planet for.
Just what car are you looking at - for Model “A” Fords (4.8 million produced), parts are not a big problem:
http://www.modelatrader.com/marketplace/marketplace.html
If you want to do a 1937 Cord, expect some problems.
As a practical matter: if there are more than 3 pople interested in a make, there is a club - find the club, and you will find out about parts availability.
Concur with the others.
I have a '66 Toronado, which is only technically an “antique”. Any part of the engine I need, from water pumps to intake valves, is available across the shelf. Mechanically I’ve had very little difficulty in getting parts. The transmission and FWD drivetrain was largely unchanged up through '78, so less-commonly-replaced parts like halfshafts and wheel bearings are also easily available.
But on other bits, I’m having a hell of a time- In particular, I need the heater assembly off the firewall. This has proven nigh impossible to find, and it’s holding up my whole project. (Thinly disguised plea for help?)
So it depends on what you need. Some “A” parts were still used on other Fords up through the mid-thirties, but many are unique. Belts are not a problem unless you’re going for period-perfect restorations. Spark plugs are somewhat more difficult, but can be had. It’s the weird things, like the manual spark-advance lever linkage, that tend to be really hard to find.
Later models tend to have at least some support from the hotrod crowd- you can still buy lots of new and aftermarket “speed” parts for even '37 flathead engines, adapters for transmissions, repop fenders, NOS taillights, you name it.
On that '37 Cord? Try and find the shaft that drives the Lycoming supercharger off the camshaft. Break that and you’d best just hire a machinist. 
Replacements just don’t exist.
And don’t get me started on the electrically-shifted gearbox.
Start scouring the junkyards in your area, find out which ones have the oldest cars still around. Also check the repair shops in town for guys near retirement, they’re the ones who’re most likely to know how to fix a problem with your car and where to find the parts you’re looking for.
There’s also parts places on the web that can help you find the parts you need. Here’s a couple to help you get started:
http://www.usedpartscentral.com/links.htm
You can also find junkyards on the web that’ll be happy to ship you the parts you need. One that seems to have a lot of parts cars is: