I have had this happen on occasion when running a long finishing path and it appears that Mach3 has locked up or lost communication with my controller. I see nothing wrong with the code and my computer has everything turned off and it is dedicated strictly to my machine. Even turning the controller on and off has no effect, it will not pick back up when using "set next line" and "run from here".
What I've had to do, without moving it, is first turn off the router. I write down the code line where it quit responding. Then write down the x,y & z coordinates. Close the file and close mach3 then power down the controller. Essentially I'm shutting down the machine leaving the router in it's last position.
I restart Mach3 and power the controller back up and hit reset. I reload the file and scroll down to the last line of code it stopped on. In Mach3 set the machine to "Off Line" and manually enter the x,y & z coordinates so it's in the same position it was when it quit, then set Mach3 back to "On line". Turn my router back on and hit "Run from here". It will continue to cut as if nothing happened.
This is the only way I've been able to resume cutting without starting all over. Believe me, I've ruined pieces trying different things and if everything is not in the correct position it WILL show one way or another. I built my machine and I've checked grounding, wiring machine set up, computer settings, etc. etc. and have not found a thing wrong anywhere. I'm not saying I didn't miss something but I've looked at everything that's been mentioned in post such as these on other forums as well. The only thing I have not done is replaced the computer since I do use a parallel port or reinstalled Mach3. It's annoying when it happens but it's something I have to live with until I decide to upgrade my computer and controller.
I hope what I've posted makes sense but the answer to your question is yes. You can resume cutting where you left off using caution.