I had this issue when importing from a flat, delimited file into SQL Server. The solution was to update the 'OutputColumnWidth' value for the offending column (from the error message). On the 'Choose a Data Source' form in the import wizard, my source was the flat file. On the leftmost pane, choose 'Advanced'. You can then set the properties of individual columns. In my case, the 'OutputColumnWidth' for most of my columns was defaulted to '50'. I simply updated it to a larger value that would not truncate the value from the flat file.
Alternatively, rather than guessing a large enough limit to a DT_STR, you can choose the DT_NTEXT SSIS type, which is the equivalent of the MSSQL nvarchar(max) or the obsolete ntext types.
A combination of updating the OutputColumnWidth and using DataType DT_WSTR worked for me. In SQL Server 2014, DT_WSTR can have an OutputColumnWidth up to 4,000 Unicode characters wide. This resulted in something similar to 40-Love's answer below.
Can you change all of these columns at once? I have a high amount of columns and I was wondering if this was possible.