Exhibit 5.1

November 14, 2025

NB Bancorp, Inc.

1063 Great Plain Avenue

Needham, Massachusetts 02492

Re: Securities Being Registered under Registration Statement on Form S-8

Ladies and Gentlemen,

We acted as counsel in connection with your filing of a Registration Statement on Form S-8 (the “Registration Statement”) related to shares of common stock (the “Shares”) of NB Bancorp, Inc. (“Needham” or the “Company”), par value $0.01 per share (“Needham common stock”), that may be offered or sold pursuant to the BankProv 401(k) Plan (the “BankProv 401(k) Plan”), which was assumed by Needham effective November 15, 2025, in connection with the Merger Transaction (as defined herein), if participants in the BankProv 401(k) Plan elect to invest in Needham common stock through the BankProv 401(k) Plan during payroll periods between November 15, 2025 and December 31, 2025, inclusive, and an indeterminate amount of plan interests issuable to eligible participants pursuant to the BankProv 401(k) Plan. Pursuant to the terms of the Agreement and Plan of Merger, dated June 5, 2025 (the “Merger Agreement”), by and among Needham, Needham Bank, 1828 MS Inc., a wholly owned subsidiary of Needham formed solely to facilitate the transaction (“Merger Sub”), Provident Bancorp, Inc. (“Provident”) and BankProv, (i) Merger Sub will merge with and into Provident (the “Merger”) on November 15, 2025 at 12:01 a.m. (Eastern Time) (the “Effective Time”), (ii) immediately after the Merger, Provident will merge with and into Needham, with Needham as the surviving entity (the “Holdco Merger”), and (iii) and immediately after the Holdco Merger, BankProv will merge with and into Needham Bank, with Needham Bank as the surviving entity (the “Bank Merger” and, together with the Merger and the Holdco Merger, the “Merger Transaction”).

We have reviewed such documents and made such examination of law as we have deemed appropriate to give the opinions set forth below. We have relied, without independent verification, on certificates of public officials and, as to matters of fact material to the opinions set forth below, on certificates of officers of the Company.

The opinion set forth below is limited to the Maryland General Corporation Law (which includes reported judicial decisions interpreting the Maryland General Corporation Law).

Based on the foregoing, we are of the opinion that the Shares, as of the filing of the Registration Statement, will be duly authorized, and when issued in accordance with the terms and conditions of the BankProv 401(k) Plan, the Shares will be validly issued, fully paid and non-assessable.

We hereby consent to the inclusion of this opinion as Exhibit 5.1 to the Registration Statement. In giving our consent, we do not admit that we are in the category of persons whose consent is required under Section 7 of the Securities Act or the rules and regulations thereunder.

Very truly yours,

/s/ Nutter, McClennen & Fish, LLP

Nutter, McClennen & Fish, LLP