Every system has a system number. When a ship enters a jp for the first time it rolls a system number to connect to. Typically, this system number is in a small band around the systems existing number, but sometimes it can vary wildly. The 'max system number' is less of a hard cap and more a variable determining how varied the system numbers you can roll are. So as you approach the max system number it becomes massively more likely that you will be connecting to existing systems rather than to new ones... You can indeed increase the max system number on the fly, or manually force a jp to connect to a new system in Spacemaster, or make more JPs.
NPRs can absolutely explore new systems.
NPRs are likely better than you at exploring, as their exploration ships do not need fuel or shore leave or maintenance. I believe they tend to only explore so far from their populated systems though.