Some time ago, I turned on Field Test Mode on my iPhone 4. I then dutifully backed up that phone for ages, restoring that backup to my new iPhone 5 when I got it last year.
Then I tried turning off Field Test Mode, which came over in the backup. And it wouldn't stay turned off.
After much searching I found a solution that, for now, appears to have solved the issue: enter the number into Notes and then paste the number into dialer, instead of typing in the number (from this helpful Apple support thread).