To keep track of your position using inertial measurement you need two types of sensor - gyroscopes, to work out which way you're pointing, and accelerometers, to work out how fast you're going.
An IMU in a smartphone would be subject to 10 m/s/s acceleration at all times under gravity, 50 m/s/s on a rollercoaster, and 100,000 m/s/s if you drop your phone on a hard surface.
Let's say we don't mind losing your location if you drop your phone, so we pick an accelerometer with a maximum range of 50 m/s/s.
Now, how accurately can we measure acceleration? The raw image setting on a fancy digital camera is 14 bits, which gives 16384 levels. CD audio is 16 bit, so it has 65,536 levels. Assume you can come up with a design that offers a 32 bit range, for a full 4 billion levels.
That means your measurements will be precise to 0.0000000116 m/s/s - pretty accurate, right?
The thing is, after 24 hours your phone will have an inaccurate estimate of its speed (0.0000000116 m/s/s * 24 hours = 0.001 m/s) and after 24 hours with that inaccurate speed estimate you'll have an inaccurate position estimate (0.001 m/s * 24 hours = 86.4 m) and the longer you leave it going, the bigger the error can get.
TLDR: You need super-precise sensors to do dead reckoning that stays accurate for long periods.
One of the secrets of useful dead reckoning systems is damping which is basically reducing your velocity toward zero (or some expected value) by a measurement slightly greater than your error. The result is if your cellphone sit's on a table you don't accumulate errors. Coupled with the occasional calibration and you end up with some surprisingly accurate systems.
Yes indeed! Of course, that doesn't reduce your position error, it only stops your position error growing while you're stationary. And you can't be stationary all the time (or if you can be, you don't need a navigation system at all!).
Inertial Navigation Systems like OxTS make [1] use GPS to get rid of this integration error that accumulates over time. For applications in vehicles you can also use the vehicle's speedometer, so you're measuring speed directly, which means fewer integration errors.
