Can I mix rice of different grains and cook them together?

Works fine as long as the cooking times are similar. White and brown rice - for instance - wouldn't work well because the brown rice takes at least twice the time that white rice does.
One of my favorite combinations is brown and wild rice. I like to make a one dish dinner with those - making a pilau of the rices with onions, celery, garlic and mushrooms, then adding sauteed chicken livers a few minutes before serving. Now I'm hungry! ;)

I do it all the time. My old rice cooker handled it really well, my new model not so much.

Agree with Monita.
When I have this problem, I cook each rice or grain separately, then combine them in main dish, where the contrast often works to make the dish more interesting.

You can, and it's a fun experiment to try. You could be scientific about it and add the rice to the mixture in stages (based on varying recommended cooking times per grain) or you could just go with the longest cooking time and expect the faster cooking grains to blend in and add a creamy consistency. Check out this article for a quinoa/rice mixing
http://www.thekitchn.com/quick-tip-try-mixing-grains-85355

As long as they are both white or both brown rice then mixing them together will be fine. But I wouldn't mix together a white and brown rice because their cooking times differ