Is it possible to achieve a negative score?
You cannot go below 0 points, so you will get a break on the penalty you have to take.

From the rules, under scoring:


  Finally, at the end of the Wall-tiling phase, check if you have any
  tiles in your floor line. For each tile in your floor line, you lose
  the number of points indicated directly above it. Adjust your
  scoring marker on your score track accordingly (however, you
  can never drop below 0 points).