Never had any dealings with NICEIC personally.
I have a customer who previously used a local electrical firm for some garden lighting, sockets etc. NICEIC approved contractors, with their own shop, nice vans etc, high price, appeared very professional.
The garden lighting failed in a short space of time, and started tripping out the RCD, taking half of the house with it. My customer couldn't get the firm to come back to fix it - several times they agreed a time and date, then didn't turn up, and eventually they stopped answering the phone.
My customer complained to the NICEIC, who did in fact visit to survey the work, agreed that it wasn't up to scratch, and forced the company to return to correct the it. I don't know what incentive was used to get them back, but I would guess that their membership was under threat.
My story seems to be unique, as I never heard of anyone else having any success in complaining to NICEIC. It seems that they would only get involved when all other options had been exhausted.