Coming back to the original question.
As pointed out by Nevrekar, you can open an account with a local bank in India. The type of account will depend on your visa type, and therein lies a pitfall if you're married and your wife wishes to avail of the same account. If you are on an employment visa and your wife on a tourist visa (or whatever else), she will not be provided with even an ATM card.
This is because (despite all the hype you hear about inflow of foreign capital) there are exchange regulations in force. On an employment visa, you get an account which allows you to remit overseas for family support etc. and an ATM card which allows you to withdraw overseas but (in theory at least) only for purposes outlined by the RBI (Reserve Bank of India). Your wife, being on another visa, will not qualify for the privileges and thus can essentially only withdraw money with your ATM card or if you give her a cheque (but even there you'll find there is some ceiling for cash withdrawals beyond which you have to produce a PAN card - this is the personal tax assessment number which you can only get if you are a resident).
You'll also find you cannot transfer money overseas online, but have to either personally submit (or if they have come to know you by fax or PDF attached to email) no fewer than 4 forms for every transaction!
Lastly, be prepared for all sorts to happen (or not) when you apply for your account. In my case, they had to submit all documents (and there are literally mountains of them) to their Bombay head office and lost them twice in the process (using their own bank courier). It took six weeks to get done what in any other country (and I've lived in seven so far) takes one hour max.