I would make sure you are using the 64bit ODBC driver if you are on 64bit Filemaker.
Which type of SQL database are you connecting to? Oracle, MS SQL, or MySQL?
FileMaker is very picky and you must have the exact ODBC driver version that FileMaker wants installed. The list of supported driver version in the link below.
If you are connecting to MS SQL ODBC with Active Directory account access to MS SQL instead of local MS SQL account, then you have to do a "Run As" to launch the FileMaker Pro or FileMaker Server and login to the account that has access to the MS SQL in order for FileMaker Pro to connect to the MS SQL ODBC to avoid the error "The account and password you entered cannot...........".
The list that FM provides is only the the database types that they tested. The drivers and FM are compatible with many more than that. I had an inquiry into FM about this when connecting to MySQL 5.4. I was able to upgrade to MySQL 5.5 but at the time my host did not support 5.6.12. The documentation suggests only 5.6.12 is supported. However, FM responded saying that is not the case and that 5.4 and 5.5 are also supported. The only use the more recent version for testing. The open source DB driver also works with MariaDB. I tested it and Actual Tech also confirmed this should be fine although they never did test it.
FM documentation is often wrong, incomplete, or misleading.
Is this an ODBC FM file you are trying to connect to?
Some time ago i went through the same thing.... ODBC connection was working, but not in filemaker. (using a plugin)
I assumed that since i was using a 64 bit OS i should install the MySQL ODBC connector 64 bit version too.
What i learned is that it will work only if you install the 32 bit driver on top of the 64 bit..... when i did that everyting started working. (Nothing will change in the creation of the ODBC)
What i noticed is that filemaker folder is located in the "C:\Program Files (x86)" which means that filemaker was a 32 bit application and i was trying to communicate throug a 64 bit driver.
I hope this helps.