Why do you think so? I think the more IIs you have, the better: it means that more schools are interested in you, so you have a higher chance of getting in
somewhere.
Now, attending all these interviews, all the expenses and missing work is another story. I don't think I would be able to attend 10 interviews even if I had 10 IIs
🙂
Paradoxically, I got my IIs (and screened secondaries, FWIW) from some of the more competitive schools on my list, but haven't heard a squeak from the less competitive schools. I'm extremely lucky that I got IIs to schools I like a lot and I think I'm a good fit for, but there is so much pressure for those 4 - now 3 - interviews
Your dozen of IIs is certainly worth celebrating! Now go for 20, Captain!
😉