I've been using the stock HH 1/8" Spectra (or similar) for suspension attachments on my homemade hammock all along, with no problems. What are you using for the hammock body? Mine is 1.9 oz. grey (somewhat coated) ripstop from Walmart. It's got a VERY sturdy feel to it. Lighter or less durable material might be the problem. Or, like others suggested, use a larger cord or run your webbing straight to the hammock.
As a side note, I originally had my straps run straight to the hammock body with no cord in between. I switched because I didn't like the way the webbing pinched the material, and the webbing itself looked like it was too stressed when looped around itself. JMO.
"Physics is the only true science. All else is stamp collecting." - J. J. Thompson