Depends on what the product is and the source of the refurbishment. I bought a Sandisk MP3 player a couple years ago refurb and it lasted me forever then I sold it on ebay and got more than I paid for the refurb (nice!). The player was refurbished by sandisk themselves, not some other place.
Apple products come with the same warranty as new and a new screen and battery, this I would definitely buy refurb, but I hate apple because I don't like the control they have over their products and their products are very overpriced for what you are getting.
Dell outlet is another good one, and I haven't found anyone who had a bad experience there with buying computers. Usually the computers there are customer returns, aka the customer that returns the computer doesn't like it so you get a hefty discount. The computers are refurb'ed by dell themselves and have the same warranty as a new one.
In general I would try to avoid refurbs of items that have built in non-replaceable batteries, because a replacement battery would probably cost you more than the device itself and its just not worth it, or between the cost of the device and a new battery you would just be better off buying the new one. I would also avoid random refurbs from ebay and amazon (and places like Gamestop who do their own refurbishment) unless you know that it was specifically refurb'ed by the original manufacturer.