The problem coincidentally wasn't with BIGINT, but with VARCHAR. The database was created with wrong collation (UTF8 at destination, ISO8859 at source), and some special characters were being incorrectly translated.
There were no display issues, but the string storage sizes were varying. And then, apparently, depending on the size of the number, independently of the string, the error was thrown.
But the truth was: the problem was only with the strings. When editing the values through TOAD or IBM Data Studio, the applications apparently fixed the collations and the problems gone away. But at my C# application, it was not happening.
I've decided to recreate the database correctly instead of fixing the data, and everything worked fine.
If only those error messages were more clearer and descriptive...
Thanks for the efforts of everyone.