I've lived in New York for about a year and a half and up until this month, I was taking cabs at least twice a day. I've had them pull the broken credit card machine excuse maybe 3 or 4 times. Obviously anecdotal, but it was so infrequent as to be a complete non-issue, and with the number of cabs on the road, it never actually caused me any problems.
In Chicago cabs are essentially required to accept credit cards if they want to pick up a fare (there's an exception for the handful of independent cab's in the city).
I don't take cabs all that often (4-5 times a month) but I have had a driver resist taking the credit card at least 20% of the time, and within that subset well over half use the 'broken machine' excuse.
Of course it always seems to work once I tell them "it's credit, or I don't pay"...
The machines are very sensitive to vapors offgassed by green dyes, but it is a psychosomatic allergy, much like RSI. (That's why cabs are yellow in color.) If you politely explain to the driver (within earshot of the machine) that you do not have cash, the machine will recover temporarily.
