As long as you're not running firmware version 1.5, you shouldn't have any trouble with your battery shutting off mid-flight. When the battery reaches the critical low level, it'll auto land at its current location. In most cases, that gives the Phantom plenty of time to make it to the ground
(or a tree/lake in some unfortunate cases).
To answer your question, you can take off on a partially charged battery if you take care to land by the time the
first battery cell reaches 3.4V. If you fly beyond that, you really need to know what you're doing because you can get into some tricky situations when trying to fly on batteries in the 3.0V - 3.3V range.
As
phantom13flyer pointed out, you should never use a battery that was last charged more than the number of days set in the "Aircraft Battery" section of DJI GO
(see my screenshot below). For example, if you have that setting set to 10 days
(the default setting) and you fully charged the battery 5 days ago, you could safely fly on it today. Or, if you used the battery for a few minutes yesterday and wanted to do a quick 5 minute flight tomorrow, that would be okay too.
View attachment 58226
The most important point to keep in mind is that your battery will start to auto discharge when the button on the battery has not been pressed for whatever "Time to Discharge" time period is set in DJI GO. After that auto discharge process has started, you should not fly with that battery again until you fully charge it.