I believe that you suspicion is correct 
From a discussion that was had a couple of months ago with the manufacturer of the PMIC chip (X-Powers), it was suggested that when the battery is completely flat, there the initial power-on current is too high, and the chip goes into self-protection mode. However, this was supposed to have been resolved in a change to the battery design, so it is not a good sign if you are suffering from that issue. If you are have the skills and willingness to tinker, the workaround is to remove the battery, and put it on a external lithium battery charger and charge it that way, and then you shouldn't have any problems, unless you let the battery get to those critically low voltages again (which typically the pinebook won't let you do).
Addendum: Firstly, please try what TLLim suggested. Secondly, I forgot to say before... someone else has suggested in the past that you try plugging and unplugging the power leading several times... It's worth a try before you take all the screws out... If the screen comes on, I'd probably try to time it with that... plug it in, pinebook LCD comes on, pull the plug, wait a second, plug it in again (while pinebook still 'on' hopefully), that sort of thing.
From a discussion that was had a couple of months ago with the manufacturer of the PMIC chip (X-Powers), it was suggested that when the battery is completely flat, there the initial power-on current is too high, and the chip goes into self-protection mode. However, this was supposed to have been resolved in a change to the battery design, so it is not a good sign if you are suffering from that issue. If you are have the skills and willingness to tinker, the workaround is to remove the battery, and put it on a external lithium battery charger and charge it that way, and then you shouldn't have any problems, unless you let the battery get to those critically low voltages again (which typically the pinebook won't let you do).Addendum: Firstly, please try what TLLim suggested. Secondly, I forgot to say before... someone else has suggested in the past that you try plugging and unplugging the power leading several times... It's worth a try before you take all the screws out... If the screen comes on, I'd probably try to time it with that... plug it in, pinebook LCD comes on, pull the plug, wait a second, plug it in again (while pinebook still 'on' hopefully), that sort of thing.