I'm not sure on health coverage, but I know it is possible to have concurrent policies for other coverages.

As long as your IL BCBS insurance is not with the Illinois Comprehensive Health Insurance Plan (I-CHIP) through the state, having multiple policies is probably permissible.

If you IL coverage IS with I-CHIP, then it doesn't matter if you take the FL plan or not, the I-CHIP coverage will end when you become eligible for your employer's plan. At least, that is how I understand I-CHIP to work. I am on the I-CHIP policy. You should also be able to get back on I-CHIP if your preexisting condition is listed and you have used up COBRA.

Usually, check policy provisions and the application. For information on which one pays first, check this link http://www.steveshorr.com/dual_coverage.htm