Yes, you have to decide which one officially overwrites the core Block. Have that one inherit the one that isn't doing the override, and have that one inherit the core one.
My_Custom_Block extends Other_Custom_Block
Other_Custom_Block extends Mage_Core_Block
Mage_Core_Block extends Whatever_Magento_Wants
Edit the config.xml files so that only My_Custom_Block is the one that is overriding the core Block.
Here's the XML you need: