Is it possible to work on extra activities (start-up project) during PhD? I am about to get my bachelor degree in computer science. I have applied PhD programs of US universities to earn doctorate degree in computer architecture subject. I kept my hopes high and applied the most of top US schools, but rejected from the most of them (mostly because my GRE, TOEFL grades are not top for these schools, I think). Hopefully I will be admitted some of the schools that I applied.

I know pursuing a PhD and conducting research is not easy, but it is a serious process that requires the one to devote himself or herself to. Luckily I like working on computer architecture and learn about new innovations, techniques, ideas proposed by researchers. I think computer architecture is a field fully open to innovation, research and development. However, at the same time I love spending time on web technologies. I have been developing new applications related with web services, mobile services, social media, content management systems etc.

Do you think would it be really possible to pursue a PhD and simultaneously work on a start-up project?

I am not looking for answers saying that it would be possible as long as you manage your time to work on them both or it is up to your advisor or program etc. I know after some point everything up to you, but I don't know the PhD experience and that's why I am asking this question here.

PS: I hope this question would help others in the sense that it is about the possibility of serious extra activities during PhD process, rather than being an personal issue.
Honestly, my instinct is "no".

Both startups and PhD advance in the same way - the massive influx of effort. And that effort is often unpredictable. Both settings have "crunch time" wherein for the next few weeks, you might as well have dropped off the face of the Earth.

Balancing that is, imo, a very tall order for a human being and will probably result in one of them suffering, falling to the side and getting triaged if you try to split the middle.

That being said, there are paths you can try, particularly if you can manage to either consult for a startup in a way that has structured limits on your time, or manage to link your dissertation to the startup's work enough that your effort counts for double. But as entirely unrelated projects? Its begging to have one fall apart.