any downside of having only one insurance company?

[Joanne Tsai]

Hi all, 

this question may be a little odd, but would appreciate everyone's input. 

Is there a downside of having all your rental properties under one insurance company? We are not familiar in this area when it comes to property insurance. Let's say if for some reason, your tenants sue you, would the premium on all of your properties go up? is it possible?

Thanks!