Typically it is recommended that RSA be used to encrypt a symmetric key, which is then used to encrypt the \"payload\".
What is the practical (or theoretical) limit
The (theoretical) limit is infinite.
For the practical limit, you'll have to make tests with your particular hardware/software implementation and compare to your requirements regarding speed.
Regarding safety, I'd say yes. Your identity (that you want hidden) is as safe as your recipient's private key's safety.