I don't think there was anything serious that didn't get reported. We never had to use anything beyond a single band-aid for non-recordable injuries that I know of and unless you want the office staff reporting every paper cut to OSHA, I don't think they were being unreasonable there.
If anything I'd worry more about the temperature of the floor or the number of hours worked, but it's hard when some people had second jobs or what have you.
It's common, when counting "days without an accident" and handing out awards and bonuses, to only reset the count for accidents of a certain severity.
After all, if you get a minor injury and you know getting treatment will cost your colleagues a $200 bonus, you might keep quiet about it and tough it out to the end of your shift.
To achieve a good culture of safety, you need to make the rewards for safety strong enough to resist the rewards for cutting corners - but you also need to make sure a guy can get a band-aid from the first aid kit or tell his boss about a trip hazard with no disincentives.
If anything I'd worry more about the temperature of the floor or the number of hours worked, but it's hard when some people had second jobs or what have you.