No. Coordination of benefits (COB) is designed to prevent that — the two plans work together to cover your costs, but you can't be paid twice or reimbursed for more than what the care actually cost.
This is due to COB rules. Here's how it works:
COB rules designate one plan as primary and one as secondary
The primary plan pays first, up to its benefit limits
The secondary plan only considers paying what remains — but specifically, it pays only what it would have paid as primary, minus what the primary already paid
Example: On a $1,000 bill where the primary paid $700, you often get nothing more from the secondary plan.