Two kinds of papers. There are two kinds of papers: self-contained ones, and those relying on published results (which I believe are the vast majority).
Checking the result. Of course, one should check carefully other's results before using them. There are several incentives to do that: become a real specialist; expand one's knowledge of concepts and techniques; find and mention a gap in the proof should that happen; get the ability to interact with more people ("I read your paper..."). So ideally, in a sense, checking a result before using it should always be the case.
Trusting peer-review. Yet, the very idea of academic peer-reviewed publications is to allow readers to locate results deemed trustable. The implied degree of trustability varies among scientific disciplines, but one would expect mathematics to have to most stringent one: a proof is either correct or it is not.
Given this, it is sometimes very tempting to use a result as a kind of "useful axiom", especially if that result has been proven with concepts very far from one's own area(s) of expertise, or if it is the culmination of several long papers: in those cases it would require a substantial amount of time, maybe even years, to personally check the results in their own right. Someone wanting to move forward quickly (or with a short-term position) may not want to go into this.
How to decide? Some cases are clear-cut (e.g. most people would accept the classification of finite simple groups), while others are borderline.
My questions on that matter are:
- Are there rules of thumb that you have come up with when deciding between checking a result, and taking it as an axiom ?
- When accepting without checking, how do you phrase it?
- Has it ever occured to you that taking a result for granted actually backfired: what happened, and what would you do differently (job interview, retraction of publication)?
EDIT (friday 7 may): many thanks to those who have replied, very interesting comments! (Also, please note that since there is no "best answer" to that kind of question I will not single out one over the others.)