Can a person have more than one basic savings account in India ?

You can open as many savings bank account as you want. I personally have five bank accounts. I used it to diversify the money across multiple banks. Also you can take benefit of the FD rates for private and PSU banks. There is no such rule for limit on how many accounts you can have. Besides almost every bank asks you for PAN account number these days which helps IT dept to keep an eye on your assets. There will never be restriction on number of bank accounts you can have because that would downplay the bank market. I have opened PPF account with IDBI and savings account with SBI. So it is definitely possible to open such account based on your requirement.

And to finally answer your question, you can have account in both SBI and IDBI.

You can have as many bank accounts - of any nature - not only savings bank. There is practically no restrictions on the number of accounts you can have in any bank or in multiple banks.
Whoever has given you the Information that you are not allowed to have more than one account has definitely misled you. I, personally, have 3 savings accounts, two FD accounts and an RD accounts across three different banks. It will not be an issue.

There is a small difference in a normal saving account and a basic saving account. Earlier Public Sector Banks used to open No Frill Accounts with zero balance to increase financial inclusion in the country. In these accounts there was a limit of only four transactions a month and the KYC norms of the banks were not that stringent as required for normal saving bank account.

In 2013, No Frill Accounts were replaced with Basic Saving Account. No minimum balance is required to be maintained in the account. Also, no account keeping charges, ATM charges etc are levied in these accounts. However, there are certain restrictions with these type of accounts like, total credits can not exceed one lakh in a year, balance cannot exceed fifty thousand rupees at any time etc.

Since these accounts enjoy some above mentioned concessions on charges, you can not open more than one Basic Saving Account in a bank as per RBI guidelines. However, since these accounts are not traced and also RBI guidelines are silent on this particular question, one can surely open another Basic Saving Account in other bank.

It must be noted that one can't even open another normal saving account in the bank, with which he/she is having a Basic Saving Account.

Hope it helps.

Yes, you can open as many bank accounts you wish to open. No bank can deny you to open multiple accounts but on the basis of your existing account they may prefer you to upgrade your account and remove the restrictions.

Regards,
Joy Joon