I recently had a storage lot that I was renting broken into, and during the investigation, I found out that the security that the owner originally claimed was in place when we rented the unit (and advertised) was nil.
There are a number of signs in and around the storage lot that say things such as "If you can read this sign, you have already been monitored" and "These premises are under 24 hour monitoring", only to find out from the owner after the theft that he has never had a security system in place at any time.
To top things off, we recently found out about a side entrance that has been left open for so long that tall grass and soil have built up for so long that the gate cannot be closed.
If we had known about the lax security, we never would have rented the unit.
Sure you can sue - anyone can, for anything. Whether you have a valid claim worth pursuing is a different issue. The first question is what your written agreement provides as far as security, or whether it provides there is no security and you waive liability. As far as the signs, the signs themselves have become an accepted practice of deterrence. Obviously, they don't prevent all crime but they may prevent some. That is not "advertising." Did you ask about security before renting? If you claim it was an important factor, be ready to answer that question. Beyond the legal issues, what did you lose and is it worth thousands in legal fees (or a hefty contingency fee) and your time to pursue? You MIGHT have some chance at a claim depending on your contract and other details.
Personal Injury Lawyer
I agree with Mr. Riddle that you need to read your documents. Often, these places will limit their legal liability or liability for damages to the amount of rent you paid for example. So the first think is look at your documents.
I do think that you may have a claim for false advertising and possibly for negligence in leaving he gate open, but will the damages you can recover make the case worthwhile?