There is no deadline. Insurance companies can rush to settlement try and get you to settle for less, or they can try to wait out the statute of limitations. Follow the advice of your attorney.
Sometimes Insurance companies try to settle quickly so they can settle for less. However, some settlements can take a long time. If it is your own insurance company then you could possibly have a bad-faith claim against them. Contacting an attorney to help you recover your settlement may be useful.
From the fact presented, i.e. the insurance company made a settlement offer, and you rejected the offer, I thus don't believe a settlement agreement was ever reached.
The more pressing question is the question - how long long do you have to file a lawsuit? This is known as the 'statute of limitations', and the lengthen of time is different in each state. For example, the general rule in Georgia is 2 years, yet there are many exceptions.
If you do reach a settlement agreement with an insurance company, the date that the insurance company delivers the check should be one of the essential terms of the settlement agreement.