No. In the case of a hard fork, there will be one block that causes the fork. This block is one that is invalid on the original chain but valid on the new chain, and is the first new block of the new chain. Because each block is related to the block before it, and that is related to the block before it, and so on and so forth, a block in the new chain that would otherwise be valid on the original chain will still be invalid on the original chain due to it having an invalid ancestor.