Hi,
I've worked at Bank of Ireland and I wouldn't touch them with a bargepole. Most of the managers did not know what they were doing. They prepared nice spreadsheets and presentations so that their department would look good on paper, but in the reality it was a big mess. Lost paperwork (deeds of ownerships), refunds (when the bank was owing money) processed at least 1 month late...
Last year, one of our clients closed his account with BoI: they agreed to send the money by direct credit to his UK bank account... and sent him a cheque! He cashed the cheque into his UK current account and proceeded to write a cheque, drawn from that account, to invest the proceeds. The BoI cheque bounced... and of course do did the cheque he had written! It took him about a fortnight of phone calls and complaint letters to eventually get the hands on his money!
I don't know how BoI managed it. Where I worked, one of the bank accounts we use for payments by BACS (direct credit) once went into overdraft by £45 million, because one of my colleagues had done a typo. The payments went off all right, despite the fact that we did not have enough funds in the account!
My guess is that if some of BoI's cheques bounce, their solvency is not held in high regard.