What would cause my door to not close even though it fits in the frame?
If the door resists being closed just as it is almost closed, then you certainly have a hinge problem. I've seen a lot of this particular problem in old houses I've lived in, where it is usually caused by excessive paint thickness on the frame and door edges at the hinge side. These faces meet before the hinges are closed, so when you force the door shut you are actually levering the hinge screws out of the wood. (Don't do that!)



You can see whether this is the problem by looking at the side of the door where the hinges are — if the door and the frame touch before the door is shut, then you have this problem. (Since you mention that there is no clearance on the hinge side, I assume this is the case.) In extreme cases, you will see the hinges lift away from the wood as you force the door shut.

You will need to fix things so that the meeting faces of the hinge plates are flush with (or above) the surface of the door and frame, rather than set below the surface (as depicted in the drawing on the right).

Planing the frame (or door) is certainly one way to do that, but I'm not a real DIYer (I just pay attention to the spaces I live in) so I don't have any experience with which to say which is the right solution. I hope I have helped illustrate the problem.