Just splitting the file up will not have the desired effect (as @A.Hersean explains in their answer).

I think what you're looking for is "Multi-party encryption"; where the secret is encrypted by multiple people's private keys, and then all of them need to participate in decrypting it. Here's a SO tread about it:

[Encryption and decryption involving 3 parties][1]

You can do a poor-man's version of this by chaining RSA encryption:

    RSA(key1, RSA(key2, RSA(key3, secret) ) )

If you want 3 people to encrypt, but only 2 of them need to be present to decrypt, then you can store 3 versions of the ciphertext:

    RSA(key1, RSA(key2, secret) )
    RSA(key2, RSA(key3, secret) )
    RSA(key1, RSA(key3, secret) )


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There are fancier versions of this called "[Secret Sharing][2]" algorithms, most notably [Shamir's Secret Sharing algorithm][3] (thanks @heinrich5991), where the secret is split up in such a way that the attacker gains little to no information unless they have all the pieces. These are still in the realm of "alternative crypto" and I don't think any are available in standard tools like openssl. You can undoubtedly find implementations, but you'll need to do some homework to decide if you trust the implementation to not be back-doored, and some dev work to integrate it into your solution.


  [1]: https://stackoverflow.com/questions/14010703/encryption-and-decryption-involving-3-parties
  [2]: https://en.wikipedia.org/wiki/Secret_sharing
  [3]: https://en.wikipedia.org/w/index.php?title=Shamir%27s_Secret_Sharing