Is it true that almost everyone who starts a PhD and sticks around long enough can get one?

From what I have seen, this is pretty much the case for PhD programs in Europe. PhD programs in Europe tend to work a bit differently than in the US. It is typically expected that you have already completed a Master's degree before starting a PhD. The PhD program itself tends to have little or no coursework requirements, and no examinations other than the thesis defense at the end.

A lot of filtering that is done in US programs through exams during the program, is instead done at the selection stage. Consequently, landing a PhD position in Europe can be more difficult than in the US. But once you got the position, it is very rare for people not to get their PhD in the end. People on occasion may drop out of the program. Much more rarely, a program may decide to drop a candidate if things are really not working out. If that happens, it typical happens in the first year. But overall in the various locations I have been, at least 80% of people who start PhDs in Europe end up with a PhD.

Some actual data:

• According to this article in the Times Higher Education, PhD completions rates in the UK are about 73% after 7 years, grow up to just over 80% eventually.
• This post by the Dutch association of Universities shows similar numbers.

But, be careful not to read what he writes out of context. The two points before this statement are:

The first is that, like a career in science, a PhD is not for everyone. It requires a peculiar mix of intelligence, discipline, creativity, rationalism, stubbornness – and sheer nerdiness. Different people have these in different measure, but a successful PhD student has a healthy dose of all.

The second is that a PhD is hard. It’s meant to be hard, not because inflicting pain is necessarily fun, nor because some scientists are ‘dementors’ (see this interesting post by Zuska on that subject), and not because your PhD is expected to solve the mysteries of the universe. It’s hard because it is an apprenticeship in science: a frustrating, triumphant, exhausting, and ultimately Darwinian career that will require everything you can muster.

In the context of "a PhD is hard" and having the right mix of "intelligence, discipline, creativity, rationalism, stubbornness" it is probably true that "almost everyone who starts a PhD and sticks around long enough ends up getting one". But, sticking around means passing classes, passing qualifiers, and passing numerous other barriers that are put there to ensure that you have the ability to finish your PhD while maintaining the quality standards of the university.

Anecdotally I know of very few people who stuck and around didn't get their PhD in the end. But, I do know of exceptions - people who stuck around and tried to get their PhD but failed.

This article from the Chronicle of Higher Education from 2013 claims that only 50% of students finish their PhD, and gives an anecdote that only 10% of one class finished.

So, in the context of what Chris Chambers writes it's more or less true, but in the broader context of people doing PhD's it's probably not.

This is a case of inverse causation. They don't eventually get PhDs because they hang around, but rather that they hang around because they are likely to eventually get their PhDs.

Those who really have no chance to graduate (which may turn out to be the case - admissions processes have imperfect prediction) will eventually be cut off funding and (need to) go elsewhere without PhDs. Those will continue to receive funding to hang around do so because they do sufficiently good work to make eventual graduation look likely.

Then again, it really depends on the country, the university's rules and even on on the research groups themselves.