The insurance company can deny your claim based on a number or factors. The following could be the reasons for your insurance claim being denied:

1. Misrepresentation of facts - if you have stated any incorrect information at any point of time from the purchase of the policy till the claim being raised then your claim will be rejected for the breach of the principle of utmost good faith.
2. Exclusions - you have claimed for something that falls under the "exclusions" category in your policy. These may include - cosmetic surgery, self inflected conditions, etc. In case of health insurance. Similarly, other types of insurance may have other exclusions. Also, insurance companies may have their special exclusions. So, while you buy your insurance policy, you must check for exclusions to avoid misunderstanding on a later stage.
3. Pre-existing diseases - in case of health insurance, if a pre-existing disease has not completed a given waiting period (mostly 2-3 years), it will not be covered. So insurance claim raised to cover such illness will be rejected.
4. Policy lapse - in case you have failed to pay your last few premiums even after several reminders from your insurer, your insurance claim will be rejected because your policy has lapsed due to non payment of premiums.

These are the major reasons for an insurance claim being rejected. These may also differ from insurer to insurer, so make sure that you go through all the documentscarefully before purchasing any policy.