See the LimitRequestLine directive in Apache's configuration. The default is 8190 so anything bigger than that has the potential to cause problems.
This directive sets the number of
bytes that will be allowed on the HTTP
The LimitRequestLine directive allows
the server administrator to reduce or
increase the limit on the allowed size
of a client's HTTP request-line. Since
the request-line consists of the HTTP
method, URI, and protocol version, the
LimitRequestLine directive places a
restriction on the length of a
request-URI allowed for a request on
the server. A server needs this value
to be large enough to hold any of its
resource names, including any
information that might be passed in
the query part of a GET request.
This directive gives the server
administrator greater control over
abnormal client request behavior,
which may be useful for avoiding some
forms of denial-of-service attacks.
I believe this is the limit of the sum of all cookies on a web page and not just an individual cookie (not positive on that though). But all the cookies for a website are transmitted using the a single request header, so if enough cookies are on a webpage and the sum exceeds the value of the LimitRequestLine directive, there will be problems.