I assume the problem is still tracing and association. So long as there is no direct link between you and the device, there is no saying if that is you or not. So grabbing a used phone via cash transaction from someone and only turning it on and using outside of your usual locations could be enough to have it being "anonymous". Then, using a vpn or other proxy that hides your network can make you very hard to find, so long as you don't associate with your real identity in any substantial form.
The danger with these technologies is not when they allow to precise track you once your general vicinity is known, that was always possible by putting some goons after you. General mass surveillance that is constantly tagged is the problem, flock, PRISM and the other stuffs.
You can be identified quite quickly by other factors if someone with access to data is looking for you. Some factors are writing style networking metadata (who do you contact) and search queries. Especially big data companies (facebook google) are tracking you even if not logged in. Some of these factors are very hard to change from one moment to another, and most people are unaware that this data is used.
The problem with mass surveillance is that even if you are aware you are actively transmitting and your data is still caught in the dragnet for example with camera doorbells and other surveillance apparatus like flock, or visiting a site with facebook ad trackers. You transmit either way by being, if you travel in public with phone or without a phone, both is an identifying datapoint.
There was a TV show in the Netherlands, called Hunted I believe. Contestants had to go on the run and avoid getting caught by the hunting team without doing anything illegal. Or staying to long in a single place, so no camping in the middle of a park (which would technically be illegal anyway).
Long story short, barely anyone got anywhere. Because simply doing an ATM transaction to get the cash already triggered the signal where they were. Not to mention the amount of government owned cameras. As well as being in proximity of anyone still online.
Complete anonymity is virtually impossible, depending on the country.