There was a bit on NPR on Saturday All Things Considered about an air-traffic controller who blew the whistle on Dallas-Fort Worth ATC's honor system which caused near-misses to be ludicrously under-reported.
When the air-traffic control center learned they were under FAA scrutiny, the reported number of near-misses went up 1700%, from 2/month to 34-- need I mention that's over one near-miss per day?
The FAA says that DFW is an isolated problem, but the whistle-blower says that the FAA is full of it, especially considering how long they blew off her attempts to alert them to the problem. I'm inclined to agree.
IIRC a "near miss" often isn't as serious as it sounds. (Though I can't find what the actual values are.)
I think 5 miles is the standard for near-miss, possibly at the same altitude or something, and you're right, 4.7 miles isn't bad at all in the scheme of things, but it makes a superb buffer zone if collision avoidance is your goal.