No, reputation can't go negative, the reason being that people who get off to a poor start shouldn't have a mountain to climb so the lower bound for rep is 1.
Interestingly the FAQ doesn't seem to cover this, but yes - No matter how much of a git someone is (even to the point of a moderator suspending them) an account will never have reputation less than 1. – voretaq7♦Dec 18 '12 at 21:01
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Considering that it takes five downvotes to counter the reputation effect of a single answer upvote, I'd figure that anyone at risk of negative rep most likely deserves it. – Miles EricksonDec 22 '12 at 14:06
People with poor start can't even ask questions anymore and don't know enough to answer anything. – Sharen EayrsDec 27 '12 at 12:27
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@SharenEayrs: Everyone who is working as a professional sysadmin should have some skills which would allow them to answer some questions and get themselves out of a question ban. – Iain♦Dec 28 '12 at 9:12