It is possible to be completely anonymous but you need to know how to do it. If you don't know how then they can trace you pretty easily. Even if you do know how, they can probably still find you if they want to expend enough resources. The only thing you can do is make it costly enough to restrict their desire to find out.
You might stand a chance if you do all the following -
1. Don't send it from a device you own.
2. Don't send it from a public device where it would be possible to track your movements to and from. (Think Tim McVeigh being caught on camera at a McDonalds. Most public places have cameras that can see and record you.)
3. Use at least 10 proxy servers chained together to restrict tracking. (Preferably ones overseas that have no legal contact with your country.)
4. Use an anonymous email website. (Again, overseas is best.)
5. Make sure you strip all information that could identify you from packets leaving your machine.
6. Encrypt all the packets between your machine and the first proxy server using a 256 bit encryption.
Of course there are still other things you can and probably should do like spoofing your MAC address.
If you do all those things they may still be able to ascertain your identity but if you move to a cave in Pakistan, you should be safe from retribution.