I also have an NRI a/c with ICICI banksince 2001 and now I requested them to close the a/c because of their excellent customer service
My experience.
1) After my initial deposit of money, I just asked them to confirm the deposit but I received a chain of emails but not the confirmation
2)This year I personally visited their Kochi branch to get an ATM card and I submitted application form & photograph and I have been promised that they can deliver the ATM card to me after 2 weeks if I can visit personaly. I agreed and after two weeks I visited their brach to collect my ATM. Customer service man told me nothing is ready and he was enquiring about my photograph. I told him that photograph was delivered to him before two weeks. He was trying to call the photographer to take another photograph. Since I was not ready to pose for a photograph he searched his table and got the photo from inside his table.
3 I requested them to send new password for internet banking (Since my a/c was blocked). I received the password but that is not working with the given user id. They are telling that my user id is changed but I don’t know the new user id. Then I called their Kochi office (ISD) to get the new userid they transferred the line to so many officers and they were trying to explain about their new products rather solving my problem. After 13 minutes I disconnected telephone. Till today I do not have my changed Userid
That is why I requested them to close my a/c