So what you are saying is that you have your own policy of insurance that you pay for over and above the insurance policy at work.  One would have to read both policies - but at a minimum your second policy - to see how it is written.  Generally peaking, though, insurance companies look at policies in a tier, so to speak, and they allocate titles to them which then in turn allocate how the claims are paid out.  They call them "primary" and "secondary" policies.  In most cases the benefits of the primary policy are paid out first and once exhausted the other policy kicks in.  If the policies cover differently that could change the generalities here.  But all claims will ask about "other insurance" so be careful here.  Fraud is not a good thing to be accused of.  Good luck.