Almost certainly no, for the same reasons that a parking deck owner is not liable for the theft of a car inside. Unless you can prove that the bicycle rack was installed improperly due to gross negligence, and the improper installation allowed the bicycle to be stolen even while locked properly, you would have no case whatsoever.
Locking a bike properly is the owner's responsibility. To prevent your next bike from being stolen, follow this advice:
- Do not use a cable lock to lock your frame.
- Use the smallest U-lock that will work (the larger the U-lock, the more easily-defeated by a bottle jack).
- Use the U-lock to lock the rear wheel to a fixed object, through your rear triangle (the one formed by the seat tube, chain stay, and seat stay). See the picture below.
- Optionally use a second U-lock, or a cable through the first U-lock, to secure your front wheel.
Sorry to hear about your bike being stolen. Good luck finding it (check Craigslist ads and/or eBay) or replacing it!
Thanks to Sheldon Brown for the image.
Update: As an addendum, if you lock up your bike at the same location daily, you can use a much heavier chain lock to lock up your bike. At work, I use that lock through the rear wheel and the rear triangle, and I have a cable that I loop through the front wheel and attach to the chain. The chain lock is prohibitively heavy to carry around regularly, but if you leave the bike at the same location frequently, you can simply leave the chain lock and cable there as a permanent fixture.
Update: My use of a chain and cable lock at work.