Remunerated consulting while being a PhD student The consulting activity, on the engineering field, would be tightly related to the research, and in fact would help to get the real-world data and reality touch to make it prosper.

Is it possible, in a big University (UC system, Harvard, MIT, Stanford or any other major college) to combine both activities?  Are there any general regulations to the matter or are they dependent of each institution policies?

In case it was possible, is there a salary cap for the total amount a PhD student can be earning?
One way or the other, you will have to disclose the activity to your university. I can't think of any obvious reason why your university would not allow you to do consulting work on the side. But in any case, talking to your adviser about it would be the correct first step -- she may be able to tell you what her expectations are if you are paid as a graduate assistant at the same time, but also what your university requires of you in terms of paperwork.