Technically, yes, there are many implementations of blockchains that do not use bitcoin. But it really depends on what you define as success and who you ask. Arvind Narayanan, a professor at Princeton, believes a private blockchain (e.g. one that does not use a cryptocurrency like bitcoin) is just a confusing name for a shared database.

If you define success as permissionless innovation – the ability for any person to be able to build applications or transact with a blockchain without needing permission from another person or organization – then yes, you need a public blockchain like those of bitcoin, ethereum, litecoin, etc.