I have a 4.0 GPA in computer science from a no name university (1001-1200 in THE) and I have 3 months worth of research experience at a good university in Europe plus one semester of a research project. Can universities like CMU, Yale, Prineton consider my application? Has anyone seen other cases like this?
While the name of your undergrad university certainly plays a role, your personal excellence is much more important. You need to show this in your CV and in your application. For example, if you are 1st author on a really good paper in your field (e.g. originating from your bachelor or master thesis) this certainly counts a lot. Glowing reference letters might also help (good is not enough).
You will be considered by any university you apply to, but for those top few places, the competition is fierce with lots of top applicants. For that reason alone, I'd say your chances are small.
While it is worth applying to such places if you have a good record and (US centric) very good letters of recommendation, you should make a broader search if you want to assure success overall. Spread some applications over interesting top fifty or so US institutions.
I know the US system is very different but what I have experienced and seen in Europe is this: you can try to get funding/apply and complete a one- or two-year master's degree at a "top" university. After that, the "top" university or a university of similar "rank" is much more likely to offer you a PhD position (assuming you did well, of course).
This is likely not directly applicable to the US, but perhaps there are options to get your foot in the door: e.g., summer course, internship, RA/TA jobs, etc.? I think your research internship is already a big plus, too.
The first year you are looking for a PhD, I would apply mostly to the programmers you are most invested in getting accepted to, but also to some research positions or masters degrees where you might get a chance to better prove yourself if you fail to get into the PhD programme you want. As others have said, publishing great papers, or having a letter of reference from someone known to the people in the department you are applying to both significantly increase your chances.
But keep in mind that competitive PhDs are competitive. It isn't just about you, it's about who else happens to apply the same year. There will always be an enormous chance element about whether the best supervisor matching your interests is even taking any PhDs the year you apply, or whether they happen to have just gotten a big grant and be taking a lot of students.
If you don't get into a PhD programme you had hoped to your first try, then you can put more time into the next tier down, or if you do get a good research position, you can put a lot of time into trying to get papers out and wind up in a stronger position for your application, or of course both.
Personally I only got accepted at the MOST competitive of the four places I applied to my first year trying. I got lucky! The professor I wanted to work with there took 8 students that year. If I'd applied the year before, he wouldn't have taken any, as he was going on sabbatical.
But having said that, I had a poor GPA from a decently-known undergraduate programme, so I didn't even apply to a PhD until I had already done a Masters and got good letters of recommendation -- in that case from references who weren't particularly well known, but from a known MSc programme.
Based on your description I'd say your odds are not good. You can tell how competitive you'll be by going to the website of your target university and searching for some PhD students' CVs.
Here is an example CV of a current student at Yale. Ignore everything from after 2019, because he enrolled at Yale in 2019. He had one publication (which won a first place prize too), several academic honours, some undergraduate teaching experience, some internship experience at Apple, and four different projects.
I don't know the specifics of course, but as you can see the competition at top universities is strong, and your competitors will be armed to the teeth.