Most people say 6 months, considering 6 months is the average time little relationships last, but it varies from person to person.
And it's not about being "sick" of each other. The end of the honeymoon period is the time when you no longer get outrageous butterflies at the thought of your partner, or think about them all day long, etc. You're a bit more settled about it. Of course, from time to time you'll get some butterflies after the honeymoon period, but not as young and pure and intense as before.