Did you do anything between it working on one purse strap and not working on the other? For example, did you change the bobbin?
If you did absolutely nothing in-between it working and not working, then that narrows down the possibilities.
What I am wondering is if you bent the needle when you were working on the strap. Take the needle out to see if it is straight. Also check the tip to make sure that the needle is intact (no broken tip). When you replace the needle, make sure that it is seated all the way up in the shaft. (Sometimes needles will slip down out of position while you are sewing, so this is a possibility too.)