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An example of an accelerometer that might be used is the one in the Wiimote, as documented in this page . This page gives the sensitivity of the accelerometer as around 300ug/sqrt(Hz).
This is almost useless unfortunately. Though it implies - given adequate filtering - that you can get a position of a meter or better over 24 hours, the very high temperature sensitivity means that even stabilising the temperature to .1C, you get a bias of up to 1mm/s^2, or 100m/s after a day, or 8600Km. (1Km in around half an hour) In real use, in a phone, without a little oven to keep the temperature stable, 3cm/s^2 or so is probably achievable, or 1Km uncertainty in around 3 minutes.
References