If we are speaking about disappearing after doing something which is particularly illegal or law breaking it will be nearly impossible. If we look at our society now, and how we operate with our lives we can say that we are not really free... there are cameras on the street which monitor us, specific companies which hold our personal information which are essential to us.
While talking about the internet, in my opinion it's impossible to be anonymous 24/7 without any trace at the moment at least. Yes, there are many programs and websites, for example; TOR, which is very popular this days and can act as a hiding spot for many users ,may have a backdoor and it surly has nodes which are being compromised by many users. There are other ways which users are using and thinking that they are anonymous, for example; I2P networks which allows the access to the materials which are stored on their own network but you would be stuffed as there isn't that much information in there as in the real "Internet" and it wouldn't be possible to access any random website.
There are also many other things which can identify you; such as MAC address which is believed to be unique and can be changed or spoofed to a slightly extent which enables the user to stay anonymous against very basic analysis which WiFi network can do against you. Actually, constantly changing your MAC address will draw even more attention, since the network administrators will see a large amount of MAC addresses associated with an access point and will investigate further.
Here's a random article about some kid using TOR thinking he was anonymous at all time:
http://www.forbes.com/sites/runasandvik/2013/12/18/harvard-student-receives-f-for-tor-failure-while-sending-anonymous-bomb-threat/#52db805d69f0
To sum it up: it just depends on who's looking for you and how much money they have. If they want to find you, they probably can.
(po prostu nie rob nic glupiego @monday)
(sources:
http://www.forbes.com/sites/runasandvik/2013/12/18/harvard-student-receives-f-for-tor-failure-while-sending-anonymous-bomb-threat/#6992fba569f0
http://grugq.github.io/
http://security.stackexchange.com/questions/29196/how-to-remain-100-anonymous-on-the-internet
http://security.stackexchange.com/questions/47262/how-to-stay-anonymous-in-public-wireless-network
http://news.nationalgeographic.com/news/2013/06/130625-edward-snowden-travel-disappear-off-grid/)
