36 years old and possibly pregnant

This might sound stupid but I need to know if this has happened to someone else. I am 36 years old and I have been trying to conceive for over 15 years. No luck. I have never been pregnant. I thought I couldn't conceive a child until a few days ago. I was supposed to start my period on Dec 28th and today is Jan 1st and I still have not started. I have taken a few pregnancy tests and they've come up positive. I'm feeling excited but at the same time skeptical because I just don't believe I can get pregnant. Has anyone at my age thought they were pregnant and come to find out it's false? I just don't want to get my hopes up because this is all I have ever wanted. I hope it's true and that I'm finally pregnant but I'm just not sure. I can't get into a Dr for a while. What could give false positives?