I see a lot of "FAILED TO CONNECT" posts here, even had my own similar issue a few minutes ago. Since I didn't see this solution posted recently, I added this thread. This is no silver bullet; there's a number of other reasons a person might encounter this problem. But maybe some lucky individual will benefit.

TS client version: 3.0.16
OS: Windows 8.1

1. I connected fine to the problem server last night. Nothing has changed since on my PC or in router.
2. I connect fine to voice.teamspeak.com, no problem.
3. Others connect fine to the problem server at the same time I cannot connect.
3. I connect fine to the problem server using tethered iPhone, with unchanged TS3 client/config/bookmark.
4. tracert makes a complete trace to the problem server.
5. The host admin has not blocked me.
6. Uninstalling and installing TS3 didn't help.

How I resolved it:
1. Navigate to %APPDATA%\TS3Client
2. Remove "cache" subdirectory
3. Restart TS3 client
4. Voila!