Yes. If you Tweet @ an account with an issue, they'll likely start to follow you and take the conversation from public (Tweets) to private (DMs) so that more sensitive information can be exchanged.

The other thing that I've seen also happen (but less often) is that you'll get a response to your Tweet with either a:
1. link to an online complaint form
2. phone number you can call
3. email address where you can send a detailed explanation of your situation

I spend a good portion of my day on Twitter, so feel free to ask away