Lizzzi, when stuck in your situation (sometimes due to my buying crappy but cheap tape and sometimes even name brand rolls act like crap), I used a SHARP x-acto knife to cut through around 2 or 3 layers, from edge to edge. You can then use the tip of the x-acto to lift up the new, thicker edge and start a new strip. This sacrifices a few inches of tape but it keeps your sanity.
I second Mississippi Mudstache’s assertion about paper over plastic in this instance though I haven't yet checked out the link provided. I used both kinds in two different businesses where we shipped products and the paper, reinforced as it was with string, was by far the stronger and easier to use product. The tape machine we had for the paper tape was a heavy old, maybe made in the 50’s, table-top affair but it never failed as long as we kept the water reservoir filled. If I was packaging things again for shipping, I’d move heaven and earth to get my hands on a similar machine.