The Waltham Savings Bank was founded in Waltham, Middlesex County, Massachusetts in 1853. The bank kept a wide range of records and those from 1853 to 1987 were donated to the Massachusetts Historical Society (MHS) in 1994.
MHS has cataloged and condensed the records into three series. Among the collection are Deposits (1853-1891) located in Series II, Customer Records (1853-1935). The catalog description for the deposits outlines that "although much of the information contained in the deposit records is duplicated in the customer account balances, the records were kept because they also contain personal information on the customer, including signatures of new account holders, place of residence, age, place of birth, and parents' and spouse's name."
I have not viewed records from this collection but there seems to be tremendous potential for these records to provide place of origin information for any Irish immigrants that banked with the Waltham Savings Bank. From 1860 to 1900, the percentage of the Irish-born population in Waltham was approx. 18-20%. Therefore, it is quite likely that a sizeable number of the bank's customers were from Ireland.
You can read the Waltham Savings Bank's collection description and how to access its records on the Massachusetts Historical Society website.
My thanks to the @Catholicgeneal twitter account for bringing this collection to my attention.
 Kristen A. Farmelant and Brenda M. Lawson. Waltham Savings Bank Records, 1853-1987 Guide to the Collection. September 2014. Massachusetts Historical Society. Available online: http://www.masshist.org/collection-guides/view/fa0132 : accessed 8 January 2017.
 Based on my own calculation from the 1860, 1870, 1880 & 1900 U.S. Federal Censuses on Ancestry.com