You can start a new game with an existing character, and your existing character may be advanced enough to play readily in the higher difficulty levels.
According to Jarulf's Guide, in Hellfire, it is standard that you can always enter any difficulty level. In Diablo, it is standard that you were never able to enter higher difficulties with low level characters.
GainTrain: You may have an old un-patched version that doesn't prevent you from joining the inappropriate difficulties.
As far as I know, the level restrictions were always enforced. This was not added in a patch.