I use a fair number of 30, 40 and even 50 year old parts when breadboarding. If the specs are suitable and the part still meets them, and you aren't designing for production, why not?
If they were kept clean and dry and at room-temperature, odds are that most of them are fine. Even the electrolytics may be OK - if there's no signs of leeakage, and they test as within tolerance and acceptable ESR, then reforming at full rated voltage may be all that's needed before you can use them.
I wouldn't bother doing much testing in advance though - the easy option is to test stuff as you need it. Clean oxidised legs with a dry green panscourer, and check parameters of passives or basic functionality for active devices before use. For pots, put low voltage DC across the track (e.g an AA battery) and scope the wiper on X1 while moving it. You'll soon spot if its noisy, and if it doesn't clean up with a couple of end-to-end turns, you'll have to decide whether its worth trying to clean them or whether to chuck them.