TENANT agrees to purchase insurance at their own expense, sufficient to protect themselves and their property from fire, theft, burglary, breakage, electrical connections etc.

More than one would be burglar has been chased down and caught by the neighbor, and most burglaries are kids stealing change out of cars. 

The same building had several burglaries, a home invasion, and a murder across the street.I try to avoid those neighborhoods now.

I understand that the property's insurance coverage does not and cannot protect my personal belonging against burglary, vandalism, fire, smoke and other perils which are covered by renters insurance.

There are a lot of burglaries - or had been around the time I bought.

our leases (California) have a paragraph about storage:"STORAGE: Lessee shall have no right of storage hereunder and the Lessor shall not be liable for any loss of property by theft or burglary or otherwise from the Premises or the building, or for any accidental damage to person or property in or about the Premises."

However, if the subject broke into the building to steal these items, you are looking at a burglary charge (a felony), rather than a simple "theft" charge.  2) One issue on a person's record can potentially be seen as an anomaly.  

Most crimes constitute Theft, Burglary, and Assault.

The following day we learned that we suffered another burglary, which resulted in a vandalized furnace, among other things.

Reports of assault, murder, and burglary/robbery are certainly things to consider and are no bueno.