No, at least not if the browser is standards compliant.
Per RFC 6265 HTTP State Management Mechanism, if the browser offers the option to disable cookies, which the user selects, the browser must not include a Cookie header in Requests, and must ignore the Set-Cookie header in response objects.
That means that not only will a server be unable to set a cookie, even if it did, your browser would not serve it up when making a request to a site on the cookies path.
RFC 6265 HTTP State Management Mechanism April 2011
7.2. User Controls
User agents SHOULD provide users with a mechanism for managing the
cookies stored in the cookie store. For example, a user agent might
let users delete all cookies received during a specified time period
or all the cookies related to a particular domain. In addition, many
user agents include a user interface element that lets users examine
the cookies stored in their cookie store.
User agents SHOULD provide users with a mechanism for disabling
cookies. When cookies are disabled, the user agent MUST NOT include
a Cookie header in outbound HTTP requests and the user agent MUST NOT
process Set-Cookie headers in inbound HTTP responses.
Some user agents provide users the option of preventing persistent
storage of cookies across sessions. When configured thusly, user
agents MUST treat all received cookies as if the persistent-flag were
set to false. Some popular user agents expose this functionality via
"private browsing" mode [Aggarwal2010].
Some user agents provide users with the ability to approve
individual writes to the cookie store. In many common usage
scenarios, these controls generate a large number of prompts.
However, some privacy- conscious users find these controls useful
source https://tools.ietf.org/html/rfc6265#section-7.2 Page 28