Is there a way to estimate the number of unfound nonces? I've heard anecdotally that the ability to find nonces is much harder than it once was. This implied that there was a fixed number of nonces in the Bitcoin system, and a large number of them have been found. 

My question is: Is there a way to estimate the number of unfound nonces?
Nonces are not 'found' or 'unfound'. A nonce is simply a little bit of data in the block header which can be changed by the miner to change the resulting hash of the block header (this is how mining works, they try lots of nonce values for their block to find one which has a hash less than some target). But because the difficulty of mining bitcoin is now so high, because there are so many powerful miners, the small nonce field can not be changed enough to find blocks any more, so miners have to change other data in the block too such as data in the coinbase transaction, or the timestamp. I believe that's what you heard.