Endless Gradle sync can indicate a missing Android SDK version

I previously noticed that if the Android SDK version requested by an Android Studio Gradle project isn't installed, Gradle sync may appear to succeed but everything will be broken. Today I found that another possible symptom is Gradle sync never finishing. A bunch of green check marks appear in the Build window, but a wheel still spins and Android Studio never proceeds to indexing. This can be fixed by installing the needed SDK version in the SDK Manager and pressing the Sync Project with Gradle Files button.