What kind of lithium ion cell did you buy? What is its physical size, brand name, and rating?
Lithium ion cells can be dangerous, and buying from eBay can be a bit of a lottery. Unless you know a lot about the vendor, you may not get what you think you are getting.
The general way of designing with batteries is you consider the power requirements of your circuit or device, things like expected supply voltage, expected current draw, expected run time, that kind of thing. Then you do a parametric search for batteries that might meet those requirements. You look at the data sheets for shortlisted batteries to see if they have discharge graphs that are capable of your application. Then, lastly, you pick a protection device that matches the battery and the use case.
You can bypass some of this if you don't want to get so fancy and just go by general design guidelines and general knowledge. For instance you don't generally want to discharge an 18650 size lithium ion cell at greater than 1 or 2 amps, though you might go up to 4 amps if you have a known high quality cell. Protection circuits attached to 18650 cells sold for hobby flashlight applications usually set the current limit somewhere in this ballpark.
As with all questions on this forum, you will most likely get better advice if you give precise and specific details of what you are trying to achieve and avoid vague generalizations.
(Also, the protection circuit is not normally supposed to cut in. The current limit is to guard against accidental short circuits that otherwise might cause damage or a fire. If there is a short circuit the protection device will isolate the battery and cut off the power.)