Collision and Comprehensive From Different Insurance Companies?

Would it make sense for me to get my collision insurance from GEICO and my comprehensive liability insurance from Allstate?  (The companies agree it doesn't make sense for me to get my homeowners and auto insurance from different companies; they just disagree on which of them should provide both.)

That's the situation we have with crop insurance and disaster payment programs.  GAO recently released a report pointing out problems because of the different rules, in particular FSA gets reports of disaster damage long after the fact.  So they recommend:

To better ensure that payments under the Supplemental Revenue Assistance Payments Program compensate farmers who experienced eligible crop losses, we recommend that the Secretary of Agriculture implement procedures so that FSA county officials are notified at the time of crop insurance claims for disaster-related losses so those officials have an opportunity to verify that crop disaster payment applicants experienced losses because of an eligible cause.

I'm sorry, but this doesn't make much sense to me.  Data flowing the other way, from FSA to the insurance companies makes a little sense--you've got one source which theoretically can propagate the data to each company.  But having the data flow from the companies to FSA is problematic.

An addendum: this FSA notice shows the problems involved with having different shares reported to RMA and FSA.