Can I lock a cursor without hiding it?

I have

 Screen.lockCursor = true;
 Screen.showCursor = true;

in my start function, but the cursor shows up locked (good!) AND hidden (bad!) when I start the game. Any ideas why this is happening or how to avoid it?

EDIT Here's the whole code:

 using UnityEngine;
 using System.Collections;
 
 public class CameraSwitcher : MonoBehaviour {
 
     public Camera masterCam;
     public Camera slaveCam;
     public GameObject slave;
     private CharacterMotor masterMot;
     private CharacterMotor slaveMot;
 
     // Use this for initialization
     void Start () {
         masterMot = GetComponent<CharacterMotor> ();
         slaveMot = slave.GetComponent<CharacterMotor> ();
         masterMot.canControl = true;
         slaveMot.canControl = false;
         masterCam.enabled = true;
         slaveCam.enabled = false;
         Screen.lockCursor = true;
         Screen.showCursor = true;
     }
     
     // Update is called once per frame
     void Update () {
         if(Input.GetButtonDown("CamSwitch")) SwitchCam();
     }
 
     void SwitchCam() {
         masterCam.enabled = !masterCam.enabled;
         slaveCam.enabled = !slaveCam.enabled;
         masterMot.canControl = !(masterMot.canControl);
         slaveMot.canControl = !(slaveMot.canControl);
         Screen.lockCursor = !Screen.lockCursor;
         Screen.showCursor = true;
     }
 }