I have insurance through my PhD program (which acts as my secondary insurance) and I'm on my father's insurance (primary)
Secondary insurance won't pay anything until the claim has gone through the primary.
We're actually in the middle of figuring things out with the primary, they might be refusing to continue covering a second session per week. The secondary has no issue with covering two sessions a week, but I don't know how it would work in terms of copay and billing and reimbursement if it still needs to be denied by my primary before my secondary is willing to cover it.
Just my primary = 80% coverage, just my secondary = $10 copay, with both together it's free. I'd have no problem with paying $10 to have a second session every week.