As someone who hires assistants seasonally, here's what I recommend (to not tick off the potential employer).
1. Double check the posting to see if it says "No Phone Calls." If so, do NOT call to check on the status of your application. That will immediately put in the hiring person's mind that you do not follow directions.
2. Double check the posting to see if it says "Do not call/email to check on the status of your application." In other words, they are saying "Don't call us, we'll call you." Once again, if you do this when the posting says not to, they will know that you do not follow directions.
3. Keep in mind that if you saw this posting, so did others. There was one season where I had 75 applicants from all across the US for just one Production Assistant job. I had so many that I had to store them in an email folder and when I finally had free time, I started sorting through them.
4. Check out the theater's website and see if they are in tech/performance and see if you can figure out who you applied to. For PA's, they apply directly to me, the PSM, and if I'm in tech/performance, anything not show related is going to be put off until the show closes, or at least until after opening. If you're applying to the Production Manager, they should be free for a follow up after a show opens. If you're applying to Human Resources, be aware that they may want to run through qualified applicants with hiring staff (PM/TD/PSM, etc) before contacting potential hires.
5. Some companies only want to send out one email. As I said in #3, I didn't have time to answer each email with a "Thanks for applying, I received your materials, we'll be in touch." Instead, once I had time, my ASM and I went through all applicants, picked those that we wanted to interview, then contacted the rest letting them know that the interview process was already full and that we would keep their resumes for the next round of hiring.
6. If 1 & 2 above do NOT apply in your case, then I recommend either of the following:
a) You can call to say "Hi, my name is xxxx and I wanted to make sure you received my resume and cover
letter for the xxxx job posting I saw on xxxx." Depending on your email situation, you can add, "I've
been having a bit of trouble with my attachments sending properly and I wanted to make sure you did
receive them." (I personally have this problem with my work email.) There is also the option that your
email got caught in their Spam filter. You could say "Sometimes my emails get caught in Spam filters,
since I use Yahoo (etc). I just wanted to make sure you did receive it."
b) You can send an email stating the same items in (a), though if your original email got caught in Spam
land, then odds are this one will too.
7. Don't be pushy. If they said they received it, they will get back to you when they are ready. If you are applying for a job and need to know by a certain date as to whether or not they are interested in you, keep that in mind and when you don't hear from them - move on.