I have never had a problem with PayPal (and yes I am verified, so they do have my bank and routing numbers).
However, one of the people I work with did have an issue and she did have some money stolen out of her bank account via PayPal. They did return her money, but it took a few days for PayPal to do that.
My advice is to go ahead and get verified, but you might want to see if you can get a free bank account that you use ONLY for PayPal, and only keep a little money in it. That way your potential losses are limited.