It sounds as if the panel is effectively being used as a junction box to a certain extent. Most (in fact I'm sure that all) of today's panels are equipped with the appropriate number of neutral and bond/ground terminals commensurate with the maximum number of circuit breaker poles.
I frequently encounter situations where people will run a home-run cable to the panel for a single-receptacle, and then again and again. The next thing you know, you have a room full of receptacles that should have either been wired in a more professional manner or at least spliced together in a separate junction box. You also end up with way too many wires in the panel. I have a feeling that this may be the situation you are describing.
I'd vote with the others and suggest a separate ground bar, move all of the ground wires to it and then have plenty of room on the original neutral bar.
