Should I Raise Rents When My Current Tenant's Lease Expires?

Good question and a great knowledge base article I will create.
From my past 10 years of owning my owner rentals and managing others, it's normally best to leave current good tenants in place without raising their rent. I actually have a few tenants that have rented from me for almost 10 years and I have never raised their rates. But, they are great tenants and never cause problems, which in the long run is cheaper then finding a new tenant.
What happens sometimes when raising rent is good paying tenants decide to move out. Lets say we raise rent by $50/month, but tenant moves out. If it takes us just 35-40 days to find new tenant(do any miscell repairs which will always need to be done), it will take you well over a year and a half to recoup the extra rent you have raised on the "new" tenant.
I have found the best thing to do is to raise rents when a tenant moves out. So when my tenant of 10 years moves out, my rental amount will go up at that time.