I've been with internet banks for a long time now, and was hit by the failure of the first internet only bank - Netbank. So the FDIC took it over and was sending the accounts to ING Direct, so in Sept 2007, I went to look for the best bank. At the time I came to the conclusion it was Everbank for me. Times have changed and rates went down and requirements went up. So it still tries to be in the top %5 of banks, but it uses a tiered system. As I don't like to keep a lot of money in low preforming places, I'm always on the bottom tier. Now I'm looking to move again when my paper checks run out, not because Everbank has been bad, I have had no problems. But because they pissed me off raising the requirement for Money Markets from $2500 to $5000, otherwise incur a fee of $8.50.
The checking account has been great for me though, I used a fee imposing ATM a couple times, and I got the fee money back. I just sent the receipts in with a deposit, and the fees were added back into my account. Overall can't say anything bad about Everbank. Their site allows for extra hard passwords, and it uses a double page entry. This might be the bank for you if you like to have a bank with more than one type of account, that all earn decent yields, and you keep enough money in the accounts to make sure you never get hit by a fee.