My experience with Bajaj Allianz has been excellent. Last year I was hospitalized for illness and was bedridden for around 5 days. I made a cashless claim to BA and it was processed without much delay. All I had to pay was the non medical expenses amounting to ~1500 rupees. They took care of the remaining 50k'ish bill.
I have a floater policy for me and my wife. For 7lacs coverage for each (exclusive so me claiming doesn't affect her insurance amount), I paid around 13 thousand. This is definitely more than ICICI and others but at the time of claims, there's never any trouble. Another thing to note is while others use third party agents for claim processing, BA has its own panel of doctors and claims are processed much faster. So a few grand extra premium doesn't hurt.
Cheers.
PS: I have no commercial interest in BA. I'm just a happy customer.