Ok, here is my experience of applying for the SBI card.
It was actually very smooth and I was getting SMS alerts constantly from SBI and Bank Bazaar
I applied thru bank bazaar to get the 1500 Amazon gift card.
Here are the timelines:
May 21, 2015: Applied for the SBI card thru bank bazaar
May 29, 2015: Guy called and collected the documents. No income statements, just last 3 credit card statements and a scan of the front of the card of a "existing highest limit card that I hold"
May 31, 2015: Got an sms saying card has been approved and would reach me in 7 days.
June 3, 2015: Got a call asking if I had a Air India fying returns account, I said no and they said they will create one for me.
June 3, 2015: Received an email from flying returns with the membership number and password.
June 4, 2015: Received an email from Bankbazaar with the Amazon.in 1500 gift voucher! sweet..
June 5, 2015: Received the card via BlueDart

I registered myself on sbicard.com, generated a PIN and made my first online trasaction.
PERFECT!

Couple of other points to note:
a) I could login to my flying returns account with the user/pass received via email
b) I applied for 2 additional cards via the sbicard.com and received both the cards without any additional documentation in 10 days.
c) I received a welcome call and I asked them to transfer the 5000 odd points I earned to my flying returns account and they were transferred in like 7 days.
Probably would have been faster if I called in directly and not have asked the welcome call guy to transfer. (He did say he will request the right department to process the transfer)
d) After my first payment, received the bonus 20K miles in 20 days.