No. In a single year you cannot contribute more than $5,000 total to all employer sponsored Dependent Care FSAs.
The best reference I have found for this is in the instructions for calculating the Child and Dependent Care Credit, in IRS publication 503. When calculating the allowed federal tax credit, you must first subtract out any employer provided Dependent Care FSA amounts first, and use the remainder to calculate any available amount for the credit. In the section "Can you claim the credit", #7 states:
If you exclude or deduct dependent care benefits provided by a dependent care benefit plan, the total amount you exclude or deduct must be less than the dollar limit for qualifying expenses (generally, $3,000 if one qualifying person was cared for or $6,000 if two or more qualifying persons were cared for). (If two or more qualifying persons were cared for, the amount you exclude or deduct will always be less than the dollar limit because the total amount you can exclude or deduct is limited to $5,000. See Reduced Dollar Limit under How To Figure the Credit, later.)
In other words, if you have two or more children (or dependents), then the federal expense limit is $6,000 minus the amount you already deducted from a Dependent Care FSA, which could be up to a maximum of $5,000. This implicitly states that it doesn't matter how many employers you've had, the max is still $5K.