As someone who was near comatose due to severely low blood sugar, passed out multiple times and had many, many other problems with it before finally getting it under control, yes it absolutely CAN get too low. 71 isn't bad though. 'Healthy' is generally considered 70-100 (mg/dL). Mine is normally somewhere between 70-80 these days and that's great. Sometimes it's a little lower when fasting, though this pregnancy has it all messed up - both low and high.
Rarely will anything above 60 or so cause major issues (though it of course varies person to person). The usual problems are temporary confusion, brain fog, sudden fatigue, dizziness, shakiness and being off balance. None of those are particularly dangerous, they're what I consider warning signs. I've never personally had a real problem until it dips below 50. And most people aren't going to experience anything truly dangerous like coma, seizures, organ failure ect. until much, much lower than that and having stayed very low for a long period of time. Those sort of problems are very rare.
If you're having symptoms, talk to your doctor (be prepared though - not all doctors are familiar with non-diabetic related low blood sugar). If not, I wouldn't worry about. Your numbers sound fine and small changes can make a big difference. I changed my diet, started eating more often, carry glucose tablets around in my purse and have just learned to watch for the warning signs. It rarely gives me any problems anymore. Good luck