Our bathroom door would not open tonight, I assumed that one of the kids locked it before closing it so I tried to poke a toothpick in the hold in the middle of the outside doorknob to unlock it, but that didn’t work this time. I took off the doorknob and the other half fell off into the bathroom. It seems the inside of the doorknob is largely made of plastic and it was obviously broken. I can’t get the bold to pull out of the door, and in trying to do so I broke the plastic even more. Now I just want to get the door open without breaking the doorjam so we can use the bathroom, but I can’t get the bolt to pull out. The doorknob is of the type with the bit that you turn vertically to lock, horizontally to unlock on the inside. I’ve put a screwdriver through the hole in the center of the inner portion and pulled on it, but it won’t move. What do I have to do?
Can you pull out the hinge pins and remove the door that way?
Nevermind, I was able to dig it out with the screwdriver with minimal harm to the door.
In the middle of the part still in the door, you should see a square hole. You need to get a big flathead screwdriver in there and turn it. That will retract the bolt.
I thought of that but they were on the inside.
Now I have to buy a new doorknob - this time, one without a lock.
I may be misunderstanding the type of lock you are references. However, the regular knob locks on bathrooms can usually be picked with two skinny knives. In the space between the door and the frame, you can see the bolt going into the frame. You can use one knife to grab the bolt and move it back a little and them hold it in place with the other and repeat. You are basically walking the bolt back using the knives in coordination. It may take a little while because the grip on the knives tends to slip and you have to repeat the process but I can usually do it in a few minutes.
Bathroom doors without locks can be a strain on families. I urge you to at least invest in a latch, especially if you have any teenage boys at home (trust me on this).
This thread is full of wrong advice.
Bed and bath door locks, for many decades, have worked on the simple pricipal of having a small hole for a small screwdriver. The other end is a simple slotted bolt. Give it a half turn and the lock on the other side also turns.
Whenever you buy a new bed-bath knob, there is a simple screwdriver “key” in the package. It looks like this http://i15.tinypic.com/7y5ieef.jpg
You are supposed to put that up on top of the door frame in the hallway, where only adults can reach it, and then it is always handy in case of emergency.
Besides children locking themselves in, anyone in the family can fall in the tub, faint, or pass out, and not be able to reach the knob.
So buy the new lock and put the “key” over the door frame.
For more of those keys see any locksmith. They may cost a dollar separately, but if you need a lot, you can buy them from Kwikset for $10 per hundred.
You seem to have missed the part in the OP that said they tried to take the doorknob off, and it fell into the inside of the bathroom.
We can do without a lock. We never used it, the custom is that if the door is closed you knock first before you come in. The door is left open the rest of the time.
Well first you need to get a potato, and then make sure the power is off. 
There is no power on a bathroom doorknob. It would create a shock hazard.
The way locksmiths remove stuck locks is with a power drill. Goes right through brass.
Bathroom doors (in residential applications) are typically inswinging, so someone in the hallway doesn’t walk into a door which is being opened.
That is based upon the older knob and spindle style mechanism, not equipped with exterior release function, which the OP tried.
See my reply to Fetchund. If you are on the outside of an inswinging door, the stop molding will preclude the use of your technique. If you’re on the opposing side, then what you describe will work, nicely, even with one knife. With one knife, you walk the bolt back and then pull or push the door to bind the bolt against the strikeplate, lather, rinse, and repeat.
True dat.
Not necessarily. Some release by merely pressing inward through the release hole to trip the privacy latch-no turning required.
Not necessarily true. If you absolutely don’t want guests snooping through your medicine cabinet, or unathorized people walking in on you, you fit your door with either a PIN activated or biometric reader to retract the latch bolt or power an electric releasing strike.
It’s not about being accidentally walked in on. It’s about getting in an argument and needing to be alone.
Wow - that’s the most verbose way of saying “whoosh” I’ve ever seen. 
I’ve had success in picking the common kind of lock myself. If there is a screw on the shaft of the knob it is possible to remove the knob and jiggle the inside - but usually the screw is on the inside of the knob. The cheap bathroom doors I’ve seen have a locking mechanism that’s really just a little slot that is turned by the cheap key already mentioned.