I wasn't able to find this information in the Hyperspec or Common Lisp: The Language (second edition). Implementation-dependent constants like LAMBDA-PARAMETERS-LIMIT and CALL-ARGUMENT-LIMIT, but not something like SYMBOL-NAME-LENGTH-LIMIT or perhaps PRINTABLE-SYMBOL-NAME-MAX-LENGTH.
The standard symbols with the longest names are UPDATE-INSTANCE-FOR-DIFFERENT-CLASS and UPDATE-INSTANCE-FOR-REDEFINED-CLASS, both 35 characters long, so I suppose that 35 could be taken as a maximum. I don't expect to ever name a symbol something longer than that, but it could matter some day.
In Common Lisp the names of symbols are strings, strings are vectors (one-dimensional arrays) and thus the length of strings is limited by array-dimension-limit.
According to CL HyperSpec http://www.lispworks.com/documentation/HyperSpec/Body/v_ar_dim.htm#array-dimension-limit array-dimension-limit is:
A positive fixnum, the exact magnitude of which is implementation-dependent, but which is not less than 1024.
Practically, SBCL reports
* array-dimension-limit
4611686018427387901
so it's not really a limit.