Exhibit 5.1

Nevro Corp.

1800 Bridge Parkway

Redwood City, CA 94065

Ladies and Gentlemen:

We have acted as counsel to Nevro Corp., a Delaware corporation (the “Company”), in connection with the issuance of $172,500,000 aggregate principal amount of the Company’s 1.75% Convertible Senior Notes due 2021 (the “Notes”), under an indenture, dated as of June 13, 2016 (the “Base Indenture”), and a first supplemental indenture, dated as of June 13, 2016 (the “Supplemental Indenture,” and together with the Base Indenture, the “Indenture”), between the Company and Wilmington Trust, National Association, as trustee, and included in a registration statement on Form S-3 under the Securities Act of 1933, as amended (the “Act”), filed with the Securities and Exchange Commission (the “Commission”) on June 6, 2016 (Registration No. 333-211864) (as amended, the “Registration Statement”), a base prospectus dated June 6, 2016 (the “Base Prospectus”) and a prospectus supplement dated June 7, 2016 (together with the Base Prospectus, the “Prospectus”), and an underwriting agreement dated June 7, 2016 among the Company and J.P. Morgan Securities LLC and Morgan Stanley & Co. LLC as representative of the several underwriters (the “Underwriting Agreement”). The Notes are convertible, in accordance with their terms and the terms of the Indenture, into cash, shares of common stock, par value $0.001 per share, of the Company (the “Common Stock”), or a combination of cash and shares of Common Stock, at the option of the Company.

This opinion is being furnished in connection with the requirements of Item 601(b)(5) of Regulation S-K under the Act, and no opinion is expressed herein as to any matter pertaining to the contents of the Registration Statement or the Prospectus, other than as expressly stated herein with respect to the issue of the Notes and the Common Stock.

June 13, 2016

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As such counsel, we have examined such matters of fact and questions of law as we have considered appropriate for purposes of this letter. With your consent, we have relied upon certificates and other assurances of officers of the Company and others as to factual matters without having independently verified such factual matters. We are opining herein as to the internal laws of the State of New York and the General Corporation Law of the State of Delaware (the “DGCL”), and we express no opinion with respect to the applicability thereto, or the effect thereon, of the laws of any other jurisdiction or, in the case of Delaware, any other laws, or as to any matters of municipal law or the laws of any local agencies within any state.

Subject to the foregoing and the other matters set forth herein, it is our opinion that, as of the date hereof:

(1) The Notes have been duly authorized by all necessary corporate action of the Company, and when the Notes have been duly executed, issued and authenticated in accordance with the terms of the Indenture and delivered against payment therefor in the circumstances contemplated by the Underwriting Agreement, the Notes will be legally valid and binding obligations of the Company enforceable against the Company in accordance with their terms.

(2) The shares of Common Stock of the Company initially issuable upon conversion of the Notes (the “Conversion Shares”) have been duly authorized by all necessary corporate action of the Company and reserved for issuance upon conversion of the Notes and will be validly issued, fully paid and non-assessable, assuming the issuance of the Conversion Shares upon the conversion of the Notes on the date hereof in accordance with the terms of the Notes and the Indenture. In rendering the foregoing opinion, we have assumed that the Company will comply with all applicable notice requirements regarding uncertificated shares provided in the DGCL.

Our opinions are subject to: (i) the effect of bankruptcy, insolvency, reorganization, preference, fraudulent transfer, moratorium or other similar laws relating to or affecting the rights and remedies of creditors; (ii) the effect of general principles of equity, whether considered in a proceeding in equity or at law (including the possible unavailability of specific performance or injunctive relief), concepts of materiality, reasonableness, good faith and fair dealing, and the discretion of the court before which a proceeding is brought; (iii) the invalidity under certain circumstances under law or court decisions of provisions providing for the indemnification of or contribution to a party with respect to a liability where such indemnification or contribution is contrary to public policy; and (iv) we express no opinion as to (a) any provision for liquidated damages, default interest, late charges, monetary penalties, make-whole premiums or other economic remedies to the extent such provisions are deemed to constitute a penalty, (b) consents to, or restrictions upon, governing law, jurisdiction, venue, arbitration, remedies, or judicial relief, (c) the waiver of rights or defenses contained in Section 4.4 of the Indenture, (d) any provision requiring the payment of attorneys’ fees, where such payment is contrary to law or public policy, (e) any provision permitting, upon acceleration of the Notes, collection of that portion of the stated principal amount thereof which might be determined to constitute unearned interest thereon, (f) advance waivers of claims, defenses, rights granted by law, or notice, opportunity for hearing, evidentiary requirements, statutes of

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limitation, trial by jury or at law, or other procedural rights, (g) waivers of broadly or vaguely stated rights, (h) provisions for exclusivity, election or cumulation of rights or remedies, (i) provisions authorizing or validating conclusive or discretionary determinations, (j) grants of setoff rights, (k) proxies, powers and trusts, (l) provisions prohibiting, restricting, or requiring consent to assignment or transfer of any right or property and (m) the severability, if invalid, of provisions to the foregoing effect.

With your consent, we have assumed (a) that the Indenture and the Notes (collectively, the “Documents”) have been duly authorized, executed and delivered by the parties thereto other than the Company, (b) that the Documents constitute legally valid and binding obligations of the parties thereto other than the Company, enforceable against each of them in accordance with their respective terms, and (c) that the status of the Documents as legally valid and binding obligations of the parties is not affected by any (i) breaches of, or defaults under, agreements or instruments, (ii) violations of statutes, rules, regulations or court or governmental orders, or (iii) failures to obtain required consents, approvals or authorizations from, or make required registrations, declarations or filings with, governmental authorities.

This opinion is for your benefit in connection with the Registration Statement and the Prospectus and may be relied upon by you and by persons entitled to rely upon it pursuant to the applicable provisions of the Act. We consent to your filing this opinion as an exhibit to the Company’s Current Report on Form 8-K dated June 13, 2016 and to the reference to our firm in the Prospectus under the heading “Legal Matters.” In giving such consent, we do not thereby admit that we are in the category of persons whose consent is required under Section 7 of the Act or the rules and regulations of the Commission thereunder.