The answer, even if you do attempt every possible combination, is no. But according to Microsoft Research, there's a way to "solve" the puzzle to within 1%, even if the board had 1000000 bulbs. The funny thing is, the algorithmic solution to the puzzle (originally built by Elwyn Berlekamp in 1960) can be used as a way to bypass brute-force computing in solving problems. The researchers at Microsoft are more interested in that whole thing, but I'm more interested in how some guy got to build this nice electronic board game as part of his day job, just for Show and Tell. Way to go, dude.
Here are the rules, in case you want to make your own home version (Phil Torrone, do you hear me?):
Update: This is also known as the Gale-Berlekamp lightbulb game—I'd hate for poor Mr. or Ms. or Dr. Gale to get left out of the credit. Also, I finally caught up on my Fringe from a few weeks back, and you commenters are totally right. Freaky. Wish I'd seen it beforehand.
Microsoft's TechFest is an annual jamboree of innovation and gadgetry from Microsoft Research, which means that while none of it is coming out as is in products in the near future, it's essentially what product development people use to add cool stuff to their actual releases. I'm here all day.!