I understand that it is acceptable to place two circuits on one breaker (assuming the load is within limits) by pig-tailing as opposed to placing two cables under the breaker terminal. However, can the same thing (pig-tailing) be done with the neutrals or grounds, when the buss bar is full and a new circuit is being added? It seems to me that to do so would require that the pig-tail be a greater gauge than the two wires to which it is connected.

Not advisable. Have you considered adding a neutral and/or ground bar? Most manufacturers offer expansion bus-bars for most of their panels.

If you start pig-tailing connections in a panel, you're headed down the wrong path, IMHO. Maybe it's time to upgrade the panel?

If the two neutrals or grounds are for the same circuit (EG, fed from the same 20A or what-not breaker that you fed the hots from), then I see no problem. If neutrals from multiple circuits are combined, the pigtail would have to be of the proper ampacity for the combined current of all the neutrals pigtailed to it.

I agree that there is likely a better solution than pigtailing, though.

If you pigtailed the neutral from 2 phases some might say you just made a multiwire circuit and you would need a 2 pole breaker under 2008. Combining neutrals from the same phase might be troubling for the AHJ too.
Most AHJs will make your life easier if you just buy the additional bus bar.
BTW
SqD and some CH breakers will take 2 wires legally. That is one reason why their panels have drilled holes for additional bus bars. Just be sure you use the additional bars for the grounds and put the neutrals on the bar array that is bonded to the service neutral in a way other than the can via the MBJ. (250.6)

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.

Thanks for all the suggestions and comments. I was not familiar with the add-on buss bar. By the way, I am not doubling up on my breakers, as I have additional room in that respect. It is on the neutral/ground buss where I am out of space.

what make of panel and breakers are they? I believe, correct me if I'm wrong, that square d breakers are designed to take upto two wires on the lower than 40amp type. chris

SqD has the neutral bars that you can cascade another strip onto. You need to free up two spaces, and the new bar has two prongs that fit into the spaces. Now you have a whole new strip.

I routinely remove these extra strips from 200A panels I install that will never see that many circuits. They are handy to have in my van.