Dealing with bankers, especially from the private sector banking space needs a lot of patience and most of the times the ability to straightaway say "no". Employees from the private sector banking space need to deliver and they tend to pile their pressure onto customers.

If you want to open a fixed deposit, you may end-up with another product
If you have a fixed deposit at a private sector bank, and you want to open another one at the bank branch, you may end-up with another product altogether. The banker knows that you already have fixed deposits, so he will try telling you that you could get much higher interest rate in alternative products. Yes, he will try selling you a Unit Linked Insurance Plan (ULIP).
Most of the subsidiaries of private sector banks run insurance companies which have ULIP products. So, some of them push for these. Yes, the returns can be higher than fixed deposits, but they are risky instruments. The returns are not guaranteed and at least a lot of the money goes into fund management, insurance payment etc.
You may end-up buying ULIPs, which a bulk of investors in the country do not understand. Be careful! Read more on ULIPS here
If you have a personal loan, there is a call to top-up that loan
If you are paying your monthly personal loan installments on time, the bank knows this is a creditworthy customer who has never defaulted. There will be a SMS or a call every month or after two months asking you whether you want to top-up the loan. You need to be patient and firm that you do not want that top-up of the personal loan, unless you are desperately cash strapped.
The reason for this is that personal loans are the most expensive loans in terms of interest outside of credit card interest. So, do not bite the bait and go for a top-up. You may be tempted to take an expensive holiday, or any other luxury spending. This is why a call from a bank to sell something can be dangerous for your financial health.
Call to pay your credit card through EMI
If you pay your credit card on time the bank gains nothing from you. For example, if you purchase a product, you may end up paying after 45 days credit period. The bank loses interest on this amount. If you end-up paying regularly, the bank gains nothing. Now let's take an example. If there is an outstanding of Rs 25,000 on your credit card and you would be paying the entire amount on the due date, there could be a call before the due date asking you whether you want to convert the same Rs 25,000 into EMIs, where you pay little amounts over a six month period.
If you agree to the same the bank will charge you interest for paying through the EMI. So, where you would have ended-up paying Rs 25,000 in one go, you may end-up paying Rs 27,000 for six months.
In case your ATM card is captured by the machine
In case you apply for a new card, you would sometimes be at pains to tell the banker that you need a simple card. You do not want any insurance on the card. A simple card may cost you Rs 250, a special one would cost you substantially higher and is the one that would most likely be sold to you. Again, if you cannot politely refuse you would end-up paying higher.
Conclusion
Bankers are just doing their job and there is absolutely nothing wrong with that. Customers are the one who should know when to say "no". If its a complicated product just refuse the same. Don't buy if you do not want.
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