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The most likely kind of data to be truncated is a string field (i.e. varchar) which is limited to 'n' characters.
If the error only shows up when using multiple threads, then one possibility is that the threads are concatenating stuff to a shared string variable, resulting in it exceeding the maximum size when the table is updated with its contents.
I may be way off beam but that's what I'd look for first.
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I would noit assume it is a threading issue. Truncated data seems to me to not be a threading issue. It might be totally coincidental that it happens when you are using threads and does not happen if you are not using threads. If it is a threads issue, then the likely cause is that you are doing something across threads that you are not supposed to do.
Regardless, it seems unlikely that anyone can help without more information and I don't know what additional information to ask for.