The meter accuracy dance is somewhere I am frequently. Though it's a waste of strips, I often test again if I see a really crazy number (like a low, since I never feel my lows). My endo has used that excuse for highs...except meters can be + or - up to 20% and 20% less than 200 is still 160.
Do you test your meter with control solution every time you open a vial of strips? If not, then you should. They make that stuff for a reason!
Some meters are supposed to be more accurate than others. The OneTouch Verio IQ and the Accu-Chek Nano are both supposed to be in the +/- 15% range, which is also not great but it's something. I personally don't find my Nano to be any more accurate than the OneTouch Ultra I had before it.
As for the other part of your question, yes, it is possible to stop feeling lows. Basically, your body burns out the mechanism for bringing your glucose back up when it's low. Part of that mechanism is adrenaline--and adrenaline causes the typical low symptoms. If your body stops responding to low blood sugars, you stop feeling them. I "feel" lows by cognitive impairment--I haven't had a true low symptom in years.
That being said, my endo has also told me to deal with lows by how I feel AND what my meter says. He told me to treat a low between 60 and 80 that I feel, and anything below 60 that I don't. True lows (meaning dangerous) are under 60. I generally treat under 70 regardless, but I'll go for something with a more simple sugar content if I'm in the 70's and feel a little funny.
One last thing--meter accuracy, even with standard meters, is +/- 20%. Erring on the high side of 20% for a reading of 60 is 72, which is borderline low, anyway. The chances are that if you're reading under 60 but you feel fine, you might be low, anyway. I'd re-test a few times to see if your meter gives you the same result again (I find that with meter inaccuracy, the meter doesn't usually throw two false numbers in a row). If your meter doesn't seem to tend to make major mistakes, I'd treat those lows.