In an action filed by the bank to recover the money transmitted by mistake, can the bank be allowed to present the accounts which it believed were responsible for the acquisition of the money?

Yes, R.A. 1405 allows the disclosure of bank deposits in cases where the money deposited is the subject matter of litigation. In an action filed by the bank to recover the money transmitted by mistake, necessarily, an inquiry into the whereabouts of the amount extends to whatever is concealed by being held or recorded in the name of the persons other than the one responsible for the illegal acquisition. (Mellon Bank, N.A. v. Magsino, G.R. No. 71479, Oct. 18, 1990)