I'm assuming these are for your home? As someone who continually struggles with paper organization, this topic has been of great interest to me.
If your bank scans checks and retains the images (I think most are because even my credit union does now), you can probably get away with discarding the bills after one year. You're probably keeping your bank statements for seven years at least. Between that and the image from the bank, you can prove that a particular bill was paid if necessary.
If you really can't part with the old system (and I understand that concept, believe me), seven years should be fine. And then shred 'em. :)
As for the bills themselves, if you wanted to keep track of past usage, Edison, Columbia Gas and Buckeye all provide electronic access to that information. I've switched to paperless billing on everything now, but it's not necessary to access the info online.