In a bunch of the cases, the problem was "hysterically bad solder joints." It wasn't so much that the joints would fail when put under pressure as that they'd fail when you looked at them funny.
Please don't melt holes into that material with a soldering iron. The plastic you are using appears to be nylon, and nylon releases highly toxic fumes when melted. In addition to the nasty smell, there are some things in there that can kill you before you notice them. I highly recommend using a drill and a small file to make access holes like that. It's faster, safer, and looks better in the end.
I'd love a citation on "some things in there that can kill you before you notice them" to update the original post with - A quick websearch doesn't find anything more than "it's carcinogenic" and the material safety data sheet doesn't call it out. (http://www.shapeways.com/rrstatic/material_docs/msds-strongf...)
Number 5 in that MSDS contains carbon monoxide which will kill you in small quantities. The "other toxic products" include HCN (hydrogen cyanide) which will kill you before you can detect it. It is produced in very small amount, but very small amounts of it are toxic. A reply to your post which is dead for some reason posted http://www.greyops.net/2010/05/advisory-burning-nylon-thread... which has a couple more compounds that are released in low concentrations when Nylon burns.
There is a massive argument in the 3d printing community about whether Nylon is safe to print with. I would use it only in an industrial/laboratory setting with decent fume extraction and filtering and safety procedures (like Shapeways has). There is a dangerous asshole who is actually selling nylon filament for printing and telling people it's safe. He has performed some tests with an HCN detector with very questionable methodology and is now making a lot of noise about how safe it is. This is why I felt compelled to make some noise about how incredibly dangerous it can be to process it in a home setting. Note that unless you subject it to very high temperatures it's not going to decompose.
I recommend a battery drill and a hand file instead of a dremel. The dremel runs at such a high speed it will melt this material instead of cutting it and you will get much uglier results.
In a bunch of the cases, the problem was "hysterically bad solder joints." It wasn't so much that the joints would fail when put under pressure as that they'd fail when you looked at them funny.