I'm not sure what you mean by 'suspicious'? Why would the bank be 'on your back'?
I bank all my earnings bar the odd bits I keep out for day to day stuff - it's far safer, I'm less tempted to spend it and makes it a lot easier to do my accounts (I buy virtually all my work stuff with my debit card, so my expenses are easy to track too). Having proper records on your bank statements and a good, provable income history will also help if and when you want to borrow money, buy property and so forth and won't hurt with the tax office either, should you ever get picked out for a going over.
If you are wondering about the bank asking you where you got the money (and it would have to be a lot - I banked nearly ?2K today in my local branch and nobody batted an eyelid) and you don't want to tell the truth, you can just say you earned it, won it at the casino or the bookies, anything you like! I've only ever been asked once where some money I was paying in was from in x number of years, I told them it was my earnings for the last few days and that was the end of it. I've never heard of any bank regularly 'questioning' a customers cash deposits, but I use LloydsTSB if that's any help.