^ I concur.
The pins on most GPIO headers are square. The holes for those pins are round. This means you are literally putting square pegs into round holes. The contact surface, at best, is just the very corners of those pins which is not nearly good enough.
You'll need solder, or a hammer header. The hammer header works because the pins have a flat, rectangular profile and have a good contact surface at either end, which is under tension.
You could possibly get your Blintk! working without solder by gently angling it to provide better contact, but that's only a very temporary solution.
Also if you use your fingernail on the corner of the top layer of your PiBow you should find a layer of white plastic film will peel away and give you a clear top.