Saving contacts to a SIM is a REALLY old-fashioned means of transferring data. It began with flip phones where you were saving only limited information.
My guess is either you have filled your SIM card (it only has a tiny amount of space), or you have categories of information in the contacts which cannot be saved to a SIM. I suggest either saving them to your computer, or the cloud if you want to transfer the information elsewhere.
Contacts dont take up much space on a phone, if your sim dies, need a new one, or switch carriers it makes it difficult to move contacts around, if you wish to move your contacts you can export them all and move them around as you please.