Animation is broken on the mobile devices after the following steps are made:
1. .fbx model with animation is imported in Unity
2. A prefab is made from the model
3. A prefab is exported to Unity package
4. A package is imported into not-empty project (such as AngryBots)

Reproduced on:
Google Nexus 5 with Android 4.4.2
Nokia Lumia 920

Reproduced with:
Unity Version 4.3.3f1 (c8ca9b6b9936)

Resolution: The quality was set too low. For example if the quality is set to fastest, then only one bone per vertex is used for the skinning. The quality can be changed in Unity editor by selecting Edit -> Project Settings -> Quality.