I prepare a script to reproduce the bug and confirmed that this PR fixed it.
https://gist.github.com/c-bata/28a6334c37f340a2ce8e54c4dee1e2bf

Before:

$ python examples/quadratic_simple.py
...
sqlalchemy.exc.DataError: (MySQLdb._exceptions.DataError) (1406, "Data too long for column 'value_json' at row 1")
[SQL: INSERT INTO trial_system_attributes (trial_id, `key`, value_json) VALUES (%s, %s, %s)]
[parameters: (38, 'cma:optimizer', '"80049528050000000000008c0a636d6165732e5f636d61948c03434d419493942981947d94288c065f6e5f64696d944b028c085f706f7073697a65944b068c035f6d75944b038c075f6d ... (2366 characters truncated) ... 00990201000000f0379026010000003037902601000000702a291301000000c0fe962601000000b025291301001400947494628c085f657073696c6f6e94473e45798ee2308c3a75622e"')]
(Background on this error at: http://sqlalche.me/e/9h9h)
...
optuna.exceptions.StorageInternalError: An exception is raised during the commit. This typically happens due to invalid data in the commit, e.g. exceeding max length.

After:

$ python examples/quadratic_simple.py 
...
Best value: 4.09659 (params: {'x': -6.36791, 'y': -36.4537})

And I also confirmed that this change has no effect to optimization results. See https://gist.github.com/c-bata/28a6334c37f340a2ce8e54c4dee1e2bf#gistcomment-3454630 for details.

By the way, I for a moment though about providing this splitting under hood of the Trial's and Study's attr manipulation methods to avoid the storage error being propagated to the user. But that might be controversial. On the other hand, we can probably catch and reraise the storage error with a more friendly one.

FYI, 2048 character limits is just an implementation detail on RDBStorage, not a specification of BaseStorage. So I think we have two options:

1. Update a specification of BaseStorage, then exporting warning messages if the length of value exceeds 2048 characters limit.
2. Add Trial's and Study's attr manipulation methods in RDBStorage without updating a specification.

I prefer the former approach, but it contains a breaking change (though it's an undefined behavior).

Let's discuss #1833 (comment) further. I'll create a separate issue for it.