They can, yes. Actually, the alternator in my aunt's car was making a high pitched whistle and she wanted it replaced, so I put a remanned one in there and it was bad. The battery light came on after about an hour of driving. I put the old one back in and it was fine until she sold the car (with the exception of the whistling noise).
With the remanufactured ones, unfortunately, you don't know how long their gonna last. Could be 10 years, could be a couple years, or could be a couple hours, lol.