Can my insurance company refuse to fix my car because I had an accident 2 week after adding full coverage to my vehicle?

First, check the policy, binder, and other documents from the insurer--see if there was any time frame or other obvious limitation which you ran afoul of (for example, did the coverage actually start or come into effect yet? It may not have, if you only just added it--the effective date might have been in the future). If there is something on the face of the policy which allows the insurer to not pay, they most likely can do this.

Even if there is nothing on the face of the policy precluding coverage in this situation, if there was insurance fraud--e.g. you added coverage, then deliberately had an accident to collect money--the insurer will not pay. Insurers are always suspicious when someone puts in a claim right after buying coverage; it may be that this is the reason they are not paying--they think you are committing fraud. However, you could challenge this--you have the right to sue the insurer to enforce their obligations under the policy, if you believe that under the plain terms of the policy, you should be paid. You do not need to take their refusal as the final word, again, unless there is something explicit in the policy which allows them to not pay under these specific circumstances (such as the effective date not yet being reached).