I had that same problem trying to use "home" machines to sew through multiple layers of heavy-weight material -- specifically, the bird's nest below. In the latest instance, I was using a brand-new Brother machine that performed just fine on other fabrics/fewer layers.

It was solved with purchase of a heavier-duty machine. The Wizard provided some guidelines to used machine purchase in another thread (heh); I found a new machine on eBay for less than $200 and so far so good.