Are odbc connections supposed to support multiple threads?

I have a java applet that originally used the mysql java connector.
The applet uses multiple threads and they all share the same database
connection. I recently started using another driver that uses odbc.
I now find that I have problems when two threads try to use the
connection at the same time. My debug output shows things like this:

thread1: CallableStatement c1 = prepareCall(...)
thread2: CallableStatement c2 = prepareCall(...)
thread1: if (c1.getMoreResults()) ...
thread2: if (c2.getMoreResults()) ...

Normally both calls to getMoreResults should return true, but with
this interleaving the second one returns false.

Is this supposed to work with myodbc? Or is there something that
you're supposed to do in order to make it work? Or was it never
expected to work? I'd appreciate pointers to existing documentation
that answers these questions.

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