I was having this same problem, and came across this question in my internet search for an answer. It turned out my problem was that I'm missing my manual, and had done some guesswork in threading the machine. I found I could view the manual here: http://www.sewingonline.co.uk/instructions/singer6202/
I had missed a little catch in the bobbin casing you're suppposed to put the bobbin thread through (pg 11 of the manual). I had also not threaded the machine exactly right (pg 12). I've now sent away for my very own paper copy of the manual, so I can avoid similar problems. What a relief to have it figured out!
Emily, November 2006