More Cells in Series Enables Full Current Charges
If you need faster charging at the upper end of the voltage range, you may have to use a different configuration, such as 2S1P, 2S2P, ... The 2S part is 2 cells in series. The nominal voltage in that case would be 3.7 x 2 = 7.4 volts. So, to charge really fast, right up to 4.2 volts, which is the typical max, you actually have to be able to charge up to the 8.4 volts (4.2 x 2), or even 3S at 12.6 volts (4.2 x 3).
Nearly-Charged Batteries Charge Slower in Constant Voltage Mode
Most likely, you have a low charging current because the constant voltage part of the charging curve has already kicked in. This actually means that your cell or battery is already mostly charged. When the battery is its deadest, and you put the battery on the charger (or use your power supply), the protection circuit built in to your cell/battery will limit the charge based upon the actual state of the battery. At first, you will be able to pump some high current into your battery, as it can take that amount of current without actually reaching a voltage that is too high, which is what it absolutely MUST avoid.
How To Test It
To prove what I'm saying, use up the energy in your battery/cell until its energy is gone and it won't give up any more. Use your multimeter to test what the voltage is on the battery. It might be as low as 3.0 volts, which is okay. Then, at that point, charge your battery with both power supplies (or chargers) and notice what current is actually being delivered to the battery/cell at that point -- it should be much more than the approx. 100mA you have already encountered.
Making A Graph Yourself To Better Understand
This should be the constant current phase of the charge. If you keep track of the charge by using a timer, and recording battery voltage, and current being delivered, at 15-minute increments, then you can end up with a nice curve that will show you exactly what is going on, and you should see the constant current phase of the charge, and also see the constant voltage part of the curve, where the current starts to go exponentially down.
Why The Current Ramps Down At End-Of-Charge
The real reason why the charge can not deliver full current during the entire charge is that it would create an overvoltage condition on the battery, because the charge delivered ultimately has to do with the delta-v, the voltage differential.
One Way Danger Plays Out
In other words, if you have unprotected cells or otherwise force a higher current when the battery voltage is already close to full, it would cause a dangerous voltage to appear on the cell, significantly over the 4.2 volt max, causing cell damage by plating of the electrodes, which would start little hairs of metal (dendrites, stalagtites, stalagmites, whiskers, ... different names are used, and different visualizations are useful), but that process which was started eventually causes the whiskers to become long enough to contact the other terminal of the battery, which causes a catastrophic short circuit, and the battery basically explodes just like a flamethrower or a rocket or some type of fireworks.
Using A Fireproof Container For Extra Safety
If you play around with these things, it is useful to put the cell/battery in a fireproof container which can be quickly covered as a precaution -- it would be good not to burn your own house down. I use a steel pot, with top, which I found used for a few bucks. And if you KNOW you're going into the dangerous areas, moving the pot + lid + experiment outside on a cement pad would be prudent.
Wisdom is to do research on this and learn what you are doing, and also to take precautions that are reasonable. Please protect yourself and your loved ones!
After reading how dangerous this can be, it should be easy for you to understand why the protections that are already in place have stopped you from shooting yourself in the foot, so to speak. Otherwise, you would have already created a dangerous situation. I am very thankful for the electrical engineers who do this for a living and work hard to protect us (and their companies from litigation), which is a win-win for everybody involved.
Visit Dave's EEVBlog For Some Great Videos Explaining These Concepts
Now that I've warned you enough, here's one place to go for good information on how to do this and some info. on how to be safe.
I enjoy watching Dave of EEVBlog, and he has some very helpful videos that you should probably watch, as a picture is worth 1,000 words. The following speak directly to your situation and are the most helpful:
This one probably shows the best constant-current and constant-voltage curves:
EEVblog #176 - Lithium Ion/Polymer Battery Charging Tutorial
EEVblog #919 - How To Charge Li-Ion/LiPo Batteries With A Power Supply
EEVblog #397 - Turnigy Accucell 6 Charger Teardown
EEVblog #811 - How The Varta 15 Minute Battery Charger Works
EEVblog #393 - LiPo Battery Discharge Testing
EEVblog #102 - DIY Constant Current Dummy Load for Power Supply and Battery Testing
Here is the list of All of Dave's EEVBlog episodes.
Search for "battery", "charger", "load", ...
Some of the older episodes seem not to be there when you click on them, but search for the title on YouTube and you'll find them.
Hope this helps.