I like ICICI Lombard amongst all other private insurance company. I previously had insurance with another company and had many troubles! I finally switched ti Lombard( based on the recommendation of a friend who is also an insurance agent) It has been more than a year and I have had a smooth sail.
The branches are well spread across the country and it is nice and accessible. (Lombard office is near my house so taht has an added advantage.) Also, I like their website. The product range is great adn one can choose from a number of different plans to suit their needs. I dont think any other general insurance company has so many options! Also, for claims, as long as we have all the documents in place and everything is foolproof from your end, is extremely fast.
Now if you only have some trouble with documents require and other processes, then in any company there shall be delay!
I like Lombard and would continue with it. Am planning to get even m vehicles insured with them now.