I just wanted to say I agree wholeheartedly about following up if you don’t get a response (and you’re serious about the professor and/or line of work).
I wrote to a professor enquiring about full time openings in her lab but she didn’t reply. I followed up after 2 weeks, and she replied almost 2 seconds later apologizing for not getting in touch sooner because she was out sick and the email then got forgotten.
In this case, there was a happy ending. The prof flew me out for an interview, and I’m still at the same lab working full time.
I was never more glad of my persistent stick-to-it-ness.