MySql has a limit of max-sizes of fields in a row of 64 kilobytes.
With varchar(255) types, this means maximum 256 fields of type text.
This is not a design flaw, it's a limit for that field type.
That said there is an easy cure in CB 1.2 RC3

until we remove that limit in next CB version.
Open file:
components/com_comprofiler/plugin/user/cb.core.xml
search for:
[code:1]varchar(255)
[/code:1]
within field type 'text' and change that one to:
[code:1]text[/code:1]
then go to CB -> tools -> check CB database -> fix.
because text is outside the row in MySql, it uses only 4 bytes instead of 256, so you can now store 30'000 such fields instead of 256... Should be enough for some time.
Don't forget to do that change in each new version until we make the fields SQL typing available in a future version.