I've been pondering this dilemma or similar.
I have a completed novel that got nowhere in the first round of submissions and is now fighting a rearguard action in the last ditch of submissions but it is getting to the point where even I have to give up.
I don't have the cash to pay a fee - am torn between 3 options. Put it away in a drawer as a curiosity and get on with second novel - which I already suspect is 'unpublishable' - or go by a recent rejection - good premise, writing ok, story does not appeal and use everything I've (hopefully) learned to write it again from scratch as a different book or alternatively, now I've put some distance between myself and it, look at it long and hard, give it the best drastic edit and rewrite I can and maybe self publish on the cheap.
I am encouraged by Colin's story.
As for re submitting - I haven't tried but I think I'd have a better idea of where to send it second time around - my first round was fairly scattershot - and if you have reason to suspect version 1 never made it past the agent's assistant's assistant - then would they even know?
Of course, looking on the bright side, could be that the first three agents on your list will all request full MS and you won't have to worry?
Sarah
<Added>I was also reading an article today about a band that has never signed to a label - have been musing on whether I could get by without an agent or a publisher...
OK - this is diasppointment talking :)