The product I'm working on uses a number of third-party products. For example, we use SSH extensively, and we use a standard SSH library - it wasn't one of those things where writing our own made any sense. We also use Java and Tomcat. Again, this is not surprising - these are used in a lot of products. Normally, this is a huge time saver and bug reducer; these products have been stressed in a huge variety of production situations and we're getting the benefit.
There's one drawback, however. These products have bugs in them (horrors!). This is one of the more frustrating things to have happen, because this isn't even code you wrote. It's code someone else wrote! Nevertheless...
You are responsible for bugs in products you use.
Your customer doesn't care (usually) what third-party products you use. The system your customer bought is yours and if it fails, then it's your problem to fix it, no matter where the actual issue lies.
So when you're testing, make sure you're testing your third-party products. And when you're fixing bugs, don't forget the ones in the products you depend on.