It is natural for any student on any course to consider whether they will navigate the full length of their course and achieve the desired goal at the end of it. With "crucial" exams starting in the United Kingdom at earlier and earlier ages, and despite failure no longer being an inherent quality of examination in today's target-driven, politically-minded environment, any individual that wishes to undertake a course of learning considers whether they are going to be good enough to finish.
Undertaking a postgraduate degree is, of course, no different, and an interesting article (via PhDweblogs.net) suggests that PhD students are more susceptible to the prospect of failure:
"In some humanities programs, only one of every three entering students goes on to earn a doctorate. No comprehensive national statistics are available, but studies suggest that the attrition rate for PhD programs is 40-50 percent."
Whilst this is by no means surprising, it does point to a consideration that can be easliy ignored when deciding to move on to postgraduate study: am I good enough and do I have the application to finish what will ultimately be a very hard grind. (On a personal note, these thoughts might have something to do with yesterday). In terms of ability, the question will, for the main part — and particularly in the UK — be answered by your ability to secure funding and a supervisor willing to take you on. The second question is something only an individual can answer.
Although the article linked above concentrates on the problem of postgraduate attrition from the perspective of department budgets, once again the individual perspective requires discussion concerning careers and the aid of a PhD in achieving desired paths.