Is it possible to be enrolled at a Masters program at one university and a PhD program in another university?
Is one allowed to do this? Or is this not possible in general?
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Maybe, but why? Different topics?
In any case, you'd have to check the rules of both universities.
As a supervisor/employer of PhD students, I would not want my student doing this. A PhD is a full-time job, requiring full-time commitment. Financing is scarce and I wouldn't want to waste it on someone who is not fully committed.
You should never, ever try anything like this without explicit written permission from the administrations of both universities, from at least the department chair level, if not the dean. I once knew someone who was enrolled in two Ph.D. programs, and it completely derailed his career when this was discovered, even though his advisor had told him it was OK.
As Dave Clarke points out in his answer, funding is a big problem. It's much worse than it might appear from the outside: it's obviously questionable to accept payment from both universities for doing full-time work, but in fact this can be a serious problem even if one (or both!) of them is not paying you in any direct way. At least in the US, there are often payments being made behind the scenes within the university. This may sound like "funny money", since it just involves moving money between accounts within the same institution, but it really makes a difference in what departments are allowed to do, so they take it very seriously. (For example, at my university, the administration charges the department tens of thousands of dollars for every enrolled graduate student, to cover tuition. Sometimes this comes from grant funding, and sometimes from general departmental funds supplied by the administration. If a student accepts an offer of admission without actually putting in the expected time or energy because of other academic commitments, then that's tantamount to a serious budget decrease. That money comes from somewhere, either a grant or the department, and whoever spent it will be very unhappy. It's not as bad as stealing, but it will be treated somewhat similarly by the university.)
Anyway, there are many different ways universities are run, so it's possible that you'll find a way to make this work. But don't try it without making sure you have written proof of official approval (keeping in mind that individual faculty members are in many cases not authorized to give approval).
It is possible. Many PhD programs don't require a masters, just an undergrad, so it is definitely possible, but as mentioned, why would you want to do that? Is it in complementary subjects that you plan to you later on? It would be awfully hard to do well in both programs. They are much more demanding than undergrad. I would suggest first doing the masters and then moving to the PhD, but it is your choice.