If the components are attractive and in good shape, you can certainly get more for your bike if you break it down and sell all the parts seperately. Shipping costs might actually be cheaper, too, if you ship a bunch of small parts rather than one big heavy bicycle. USPS Priority Mail is dirt cheap for small packages.
Do some ebay research and add it up - watch other auctions for the same parts you have, and see what they go for. If you've got 1980's Campagnolo C-Record with Delta brakes, then you definitely want to sell part-by-part. But if it's five year old Shimano Altus or something, you might as well leave it on the bike.
I sold a track bike a number of years ago and got more for the cranks (Sugino 75) than I did for the frameset. And the cranks were a lot cheaper/easier to ship, too.