Up to 2000 measurements are stored on the Pulse device without offloading them to the phone. If you get to 2000 and take another measurement the first measurement you took will be overwritten.
Whenever you bring a Pulse within range of the phone that it is paired with, the phone will automatically download all the measurements stored on the Pulse device that the phone does not currently have a copy of. This means that, as long as you let the Pulse device sync with your app at least once every 2000 measures, the number of measures you can store is limited by the memory size of your phone.
Remember also that you can export the measurements from the app into a CSV file that can be easily opened by excel or similar.