I might be experiencing the same problem.
I installed the plugin (version 0.9.4) and successfully linked the Dropbox account.
When I click the Start Backup button under Backup Now I get the following message:
Your backup has been scheduled and is waiting for WordPress to start it. This could take a few minutes, so now is a good time to go and grab a cup of coffee.
However nothing seems to happen, even after an hour no files are visible in Dropbox.
The plugin What's in Cron tells me this:
Apr 11, 2012 @ 9:11 (1334135482) One-off event execute_instant_drobox_backup
Apr 13, 2012 @ 2:00 (1334282400) Once Daily execute_periodic_drobox_backup
I started a manual backup (using Backup Now) yesterday at 9:11 UTC, which is visible in the Cron list, but it didn't result in any backuping being done. The nightly/daily backup at 2:00 also didn't run.
A new Cron for tonight (it is currently Apr 12, 2012) is planned, but a new manual backup I started about 30 minutes ago is not visible.