fizz57 et al:
When using Li-Ion (as well LiPo) cells it is important to have each cell voltage charged (or at least balanced) separately.
This means that when nominal cell voltage is 3.7V and you have e.g. 3 cells in SERIES to get a 11.1V potential, it won't do the job to charge them with a single 12.9V supply UNLESS each potential is balanced (i.e. 3 x 4.3V charge voltage applied to each single cell level).
If cells are in parallel, then it is of utmost importance that both cells have the same charge/discharge curve, otherwise current balancing is not possible between the cells, which may lead to under- or overvoltage discharge or overcurrent draw of single cells.
Thats at least how I learned it when designing large-ish power applications for Li-Ion batteries. And they still work fine, no issues, no interruptions, no anything.