Is it technically feasible to attempt finding a wallet's private key? I thought it was practically impossible to do it with today's computers. I have already lost a few keys and know I have coins on blockchains that I can no longer spend.
It depends. The strength of your private key is directly related to its randomness.
I'd say the standard client generally would provide enough randomness so for all intents and purposes it is infeasible.
On the other hand if you did a sha256 hash on some input for your private key (i.e brain wallet etc) it will greatly depend on how easy it is to guess.
Just look at the sha256 of an empty string -> e3b0c44298fc1c149afbf4c8996fb92427ae41e4649b934ca495991b7852b855 and its corresponding address 1HZwkjkeaoZfTSaJxDw6aKkxp45agDiEzN
If an early miner had compromised random number generation you might be able to exploit that but this issue is universal to all addresses.