Getting Android error message "App not installed"

I'm having trouble installing some APKs on my ASUS tablet. To avoid confusion with other posts, and to help clarify the specific problem, let me eliminate some things right away.

I do have the Security setting to accept apps from "Unknown sources". I can install many APKs without a problem; I have hundreds on the tablet. I do not have a rooted device, and have never made any system-level changes to my software. And the very same apps that fail on my tablet will install on my Galaxy S4 phone, as well as a different (older) tablet.

It would appear that there is something "intervening" during the install process. When I tap on the APK file (which I have tried from both the internal, as well as external, storage), it begins as normal: I see the Installer come up, I am presented with the Privacy and Device Access parameters, and I tap on "Install". Within a second or two, I get a pop-up saying "App not installed", and the only option is "Done".

The APKs that fail do not come from a single source. Some have come from my Amazon Android library; some have come from well-known Android game sites (e.g., the Humble Bundle site). I am not able to nail down which will, and which won't, install (the list would be too long). However, whenever there is one that will not install on the ASUS tablet, it will NEVER install. (I.e., the error is reproducible.)

And to avoid a few other rabbit trails, I am not re-installing any of these apps; they are all new, with no naming conflicts with existing apps. Some of the apps are installed by sideloading (i.e., copying the APK to the device and clicking on it); the Amazon apps that fail have used the standard Amazon Install process (i.e., the Amazon app on my tablet). I use Lookout to scan my apps, and there are no problems. And all of my devices are running Android 4 -- they range from 4.1 to 4.4.4. All are kept as up-to-date as possible.

Can someone give me an idea of what is happening, what causes the "App not installed" message, and why all of my other devices will install the same app?

Thanks.