This really is a myth. I broke a frame back in the late 80's and had a machinist friend repair it. This is a guy who welds exotic metals, makes pressure vessels and fabricates lots of one off equipment for scientific experiments. He knows his stuff. Even he was amazed at how hard it was to weld the thin steel of a bicycle frame. He said it would be incredibly easy to burn through.

If you want to see a real life example of a repaired bike at a local blacksmith's near Outer Mongolia, look here.