How old is too old for a PhD?

I would imagine most institutions would consider it discriminatory to judge on age, and in some places it would just be illegal. The real consideration is always whether the prospective candidate has a reasonable chance of success. Having appropriate education, or a reasonable substitute is of course the main criterion, though demonstrating research-level ability always helps (research publications etc.).

Personally I have seen PhD candidates of all ages. There's some skew to the distribution across disciplines (business and history for example often have higher proportions of older doctoral candidates than, say, mathematics), but this seems to be more a socio-cultural thing than any institutional influence.

At the university where I work, there are no limitations or guidelines on appropriate age ranges for a Ph.D., and I'm sure this is true throught the US since age discrmination would be illegal. In practice, we see the opposite problem regarding age: the application rate in mathematics is very low after the mid-20's, although we would be happy to consider older candidates. It may be that they just aren't interested in applying (if you have a family or are used to earning a high salary, then going back to school may be difficult or unappealing; furthermore, everyone is exposed to enough math in childhood that perhaps relatively few people first discover a fascination with it at an older age). However, I fear that there are people who would really like to go back and get a Ph.D., but who do not apply, because they believe they are too old to do mathematics research or because they do not believe they will be admitted. That would be sad, since I've known several extremely successful mathematicians who entered grad school well beyond the typical age.

This is a slightly different view of some issues raised in a recent answer.

That answer stated some of the disadvantages of being a mature PhD student, but missed several major advantages that mature students can have, especially if they have worked in their field of study.

I worked in the computer industry from 1970 to 2002, when I left to become a CS graduate student, completing my PhD in 2009.

Long hours. The article asked "Are you ready to pull all-nighters and push yourself to the limit (mentally and physically)?". I was definitely not ready to pull all-nighters, and never needed to. I had spent decades on projects far more complicated than one student's studies, and knew better than to leave a hard-deadline task until just before the deadline.

Over-long working hours lead to mistakes and reduced productivity. It is much more efficient to mix work with plenty of sleep and a reasonable amount of exercise and relaxation. Mature students have had more time to get work-life balance under control than recent graduates.

Not well paid. The article seemed to assume financial dependence on the PhD program. That is true for most recent graduates. Mature students, especially after successful careers in a technical field, may have other financial resources. I supported myself and paid my tuition out of my investment income while living in a house I owned.

Background knowledge. This is the area where a mature student may have the biggest advantage. There is an immense difference between passing an undergraduate course in a topic and spending several years of one's working life living and breathing it.

Staying employable in the computer industry for decades requires continuous study. I had already read some of the assigned papers for seminar courses, and earlier editions of a couple of course textbooks. Studying was easier in college because I could get advice on what to read, saving me the effort of working out what I was going to need to know next year.

After the PhD. In my case I took a year off to celebrate after the PhD, and enjoyed it so much I retired. I can still use my skills answering questions on StackOverflow, helping a college robotics team, and participating in open source software development. That is an option I would not have had thirty years earlier.

If I had looked for paid work, I would not have considered jobs as a postdoc or similar. That would ignore almost all of my resume. I was more likely to return to industry, because I prefer a pure technical path that the academic world does not seem to offer.