Mike,
After hours of thorough testing on Chrome I've come up with a few consistent results. Each test added or subtracted a single scale dependency setting and then previewed the results in the ESRI "Basic Viewer" Web App template. There was no cache-clearing between each of these tests.
Web Maps designed in ArcGIS Online that have multiple layers will function properly in ESRI's out-of-the-box Web Apps as long as only one layer has a scale dependency set. After the second layer is given a scale dependency there can be issues. I'd like to be able to provide a list of all of the issues but they can be so random that even a test performed the same way twice may produce different results. The most obvious problem would be missing layers. These could be ones with scale dependencies set or they might not have even been altered in any way at all in the table of contents. Of the missing layers that did have scales set, if you zoom to an extreme (far in or far out) you may be able to get it working again but others may stop working. Additional issues included graphical lag in panning and zooming (features appeared to hang in the air then snap back to the ground) and what appeared to be a delay in settings by up to two tests. What I mean by that is a setting toggled in Test #3 may not appear to work until Test #5. This issue is harder to pin down and may only be a misperception of the more random results.
We would like to know if this is something we should to take to the Tech Support level or just keep an eye on the forums?
Thanks,