Hello. It really depends on where you were in your cycle when you first began taking the pill. If you started taking the pill on the first day of your period you would have been protected against pregnancy from then on. Otherwise, you are supposed to wait 2 weeks or so. Most doctors will say to wait one month before trusting the pill (i.e. not using a barrier from of protection, such as condoms).
The three months you are thinking of is how long it may take your body to get used to the extra hormones. You may be protected from day 1 (if you started as I mentioned above), but it may take your body awhile to accept the change. This just means the side-effects, such as nausea and breast tenderness. The side-effects should gradually diminish and by month 3 shoudl hopefully be gone. If not, and you don't want to put up with them, you can always see the doctor and get a lower-dose of the pill. I know you didn't ask this. I just thought you might like to know.
In answer to your question..
After the month you should be over 99% protected by the pill.. if you have been taking it on time and every day. So there is always a VERY small chance of pregnancy no matter how long you have been taking the pill and where you are in your cycle.
One of the purposes of the pill is to supress ovulation. i.e. an egg is not released to be fertilised, so you cannot get pregnant.
If you take your pills correctly the whole month you should be protected throughout the next month, and the months to follow that.
This includes during the first week of pills after you start a new packet.
I hope your question is fully answered somewhere in all of that.
You shouldn't worry.