1st and 2nd generation. I've had the older one for 14 months and the newer one for 2 1/2 months and I've never experienced that sort of problem. I also have 5 other iPods that have a very good track record. Have you tried syncing or charging it again? If it doesn't work, definitely return it.
As far as reliability, with any device, there are going to be a few bad apples (pun intended). Of course, you don't want to be the one that buys it, but what can you do? Return it.
I got an Ipod touch 2 gen (16gig) on nov 25th & well later today it would not turn on... not a thing, this morning it worked fine & was 90% charged. I got it from amazon so it was with-in the 30 return/exchange time.
How common is this????
I am definitely going to get the Apple extended warranty, since this is my 1st MP3 player, it is not a good 1st start.
I did back-up everything.. so that is good.

Chowhound
Comic Vine
GameFAQs
GameSpot
Giant Bomb
TechRepublic