Does proxy servers and IP address altering/hiding software available to the public truly work ?

I have always wondered about people getting caught by the local police for cyber-bulling or stalking (even mean/spicy comments on Facebook to an unwelcome party)
Why didn’t they hide/change their IP addresses or use a proxy server to remain anonymous,do their evil deed and stay out of trouble?
or is it impossible to hide your IP address from the local law enforcement authorities and websites as Facebook,Myspace,Yahoo etc,,etc…

I understand individual making terrorist threads or threatening people’s lives/safety being unable to hide from the federal authorities as that would be a serious crime.