Arsen is correct that this is the way to go into Vibrate mode, however one thing to watch out for is:
Setting phone to modes other than silent or normal can overridden by turning the volume down too far.
To repeat this issue do this:
- Set phone to any mode other than normal or silent, vibrate for instance.
- Turn down the volume using the side volume buttons until the phone vibrates, this is a different way to turn the phone into silent mode. (You will see the silent mode indicator shown in the image below pop up.)
- Now you will notice the phone is set to 'Silent Mode' and no longer in "Vibrate Mode."
- Turn the volume up one notch, the phone will vibrate once and you will be back to "Normal Mode" and you will be hearing notifications.
The phone does not seem to remember what mode you had set prior to going into silent mode with the volume keys. I wish they would simply get rid of the volume key 'silent mode' switch.. it is simply unnecessary (easily achieved by tapping notification icon on main menu bar)
I learned this on a major motion picture set, where phones must always be quiet.. lest you ruin a very expensive stunt or piss off a very expensive actor or director... moments before my phone rang.. 
This may not be the same issue you are experiencing, but it may be related, so I thought I should point it out. 
