I just tested the base library included with 0016 and it worked perfectly. Using the sample sketch I posted at the top of this topic I was able to make 50 consecutive connections to my web server and retrieve the page without a single failure. I don't see any problems.
My test environment is OSX, a Duemilanove with a ATmega 328, and an official ethernet shield.
If you're messing around with the source files for the library, be sure you delete the .o files otherwise the library code won't get recompiled and you'll be using an old (and probably incompatible) version.