Simplenews newsletters not being sent out when Mime Mail is enabled

I'm having a problem on one particular client's site, which has Simplenews and Mime Mail installed. The newsletters are failing to send if Mime Mail is enabled (they send fine if I disable the module, so the problem appears to be with Mime Mail rather than Simplenews). This is a first, as I have both modules installed on a number of other sites, none of which have any problems sending mail, but this one is with a different host.

Drupal's log shows an error message, but it's rather vague and unhelpful ("Outgoing email failed"), though it does note that it was sent via Mime Mail, so again, the problem does not appear to be with Simplenews.

I can't seem to find the web server's error log (have e-mailed the host about that), so I'm not sure what to check next in the meantime, in order to troubleshoot this. But I'm guessing the problem has something to do with the web server or e-mail configuration at this particular host, since Mime Mail works perfectly well with Simplenews on every other site I've installed it on.

Has anyone encountered something like this, or does anyone have any suggestions on what variables in a hosting environment might cause Mime Mail to fail, so that I can know what questions to ask the host? Or, failing that, any other ideas as to what else I could check in order to troubleshoot the problem further?

If it helps any, the site is running the current versions of Drupal core and all its modules, and the server is some flavour of UNIX, but I don't know which offhand. It's a shared hosting setup, and while I'm used to working with those, this one seems to have a few more restrictions than others I've dealt with.