At the time, I think it was literally one per pageload to get the balance of an address, and it occurred on about the third refresh in 20 seconds (I had recently updated the file and was waiting for it to propagate to my hosting service). I think the maximum it could have been was 5 API calls per pageload, but it only caused an error on one.
Currently the minimum amount of requests on a pageload is 5, with a max of about 10 if a lot of stuff goes wrong, and if I remember correctly I've only seen the error once more, and pretty close in time to when i saw the original one.
I do a lot of calls in a short period of time to manually test things, but it's still a relatively infrequent thing, and I usually only do about 4 pageloads before i see the next bug to work on.
There is only one point in my program that this could cause an error (when i'm using the balance of an address for some math), so I should be able to catch that and abort gracefully if this shows up again, but I get that other people's implementations may be less flexible and have a lot more riding on them.
I mostly use blockchain API because I'm not hosting anything myself, and I don't want to have a lot of setup time to install daemons if I have to change hosts.
Before starting real web development with bitcoin I would have never thought that paying for enterprise-level garauntees for a reliable API was worth it, but now giving a portion of any income to blockchain for garaunteed uptime doesn't sound so bad.