https://wiki.unrealengine.com/Destructible_Troubleshooting_Guide
http://timhobsonue4.snappages.com/project-dm-showcase
These two links should help you sort most common destructible issues.
From your short description, I would guess that you’ve not set the Support Depth to 1 for your Destructible, but have enabled Impact Damage. Without Support Depth set to 1 or higher (higher if you’re using a destructible from PhysXLab with multiple depths).
If you’re still having issues after checking the links above please post your settings you’re using in your Destructible Mesh Editor since this would be really helpful to make sure you’ve set it up properly.