First company is X and second company is Y
X verifies the entire claim and bills and then pays upto your allowed limit. Y needs settlement letter from X as you have submitted original docs there. Based on the settlement letter Y will pay only the remaining amount. Y may ask for documents/xerox for their verification and you need to get this clarified before submitting the claim to X.
Another problem might be last date of submission of claim at Y, as you have to wait for X to provide you the settlement letter. In this case you can pre-inform Y about the delay. Keep every communication in mail or print, do not believe assurances given on call.
I have not claimed from two companies but this entire process was explained to me by our HR mediclaim SPOC.