Jessie, if you run the thread through a disk of beeswax before you use it that will help too. Just pull it through the wax after you have threaded the needle. Be sure to pull it through the needle quite a ways and check it after a few stitches. Mine pulls up sometimes but having eyeballs I can see when it does that!
Ah! I think I know your problem! The eye of the needle has worn the thread and has broken it (if I understand that you were using a doubled thread, yes?) You might need to change your needle if it keeps happening, I usually toss them if they do that as it means they have a rough bit inside the eye from manufacture and aren't worth persisting with! You can rethread both the threads but you will have to be careful when pulling your stitches that you don't keep pulling the thread off the needle! It will probably be easier for you (with no eye balls ) to simple start a new thread. Of you go back a few stitches and resew a little you should be able to then snip off the old thread.
I hate when that happens too, but I discovered a new needle threader from JoAnn Fabrics that runs about $14.99. You place the thread through a slot and put the needle through the hole and press the leaver and presto, it's threaded. I use the smallest needle possible and never thought that this would work but even wih the thread doubled, it goes through. Now I don't care if my thread comes out because it's easy to rethread it. I bought it on line but you can get it at a large JoAnn Fabric store also.
Good luck. Audrey