It had nothing to do with whether the bank could pay higher interest to her customers. It had everything to do with not having the liquidity to give customers the money they had in their accounts when a large number of customers came looking for their money at the same time.Yes, due to bank is unable to pay higher interest to her clients.
The reason it didn't have sufficient liquidity to repay it's customers' deposits was because the recent hike in interest rates devalued the bonds they had invested and relied on. They therefore sold them at a loss, and the bank collapsed as a result, unable to fill the hole through raising capital.
So higher interest rates were part of the reason for the collapse, but most certainly not for the reason you suggest.