I don't think this if/else construction is needed. subprocess.Popen's stderr argument defaults to None, so you can just call bitcoind_processes[i] = subprocess.Popen(args, stderr=stderr)

yuck! raising an exception in expectation of catching it in the very next line.

I think try/except/else is a much cleaner construction, as in:

try:
    thing_we_expect_to_fail()
except ExceptionType as e:
    do_some_asserts_on_the_exception_raised()
else:
    # oops, we were expecting an exception
    assert False, "we were expecting an exception!"

Consider opening the file using the with log_stderr as... form. That means the file will be closed automatically (even if an exception is raised on the way), and you won't have to close it yourself with the finally branch at the bottom.

minor nit: assert is a statement, not a function. This should be:

assert 'bitcoind exited' in str(e)

Nit: No need to stop the node here. This is done by the test framework.

It does need to be stopped, or this function would not be side effect free.

EDIT: Ah no it does not, as if it passes here, the test is finished, which would close the node anyway.

Right this line is only executed in exceptional circumstances, in which case the whole frameworks shuts down.