It generally takes 12 days before your body resets it's relevant receptors for caffeine consumption. E.g if you drink three coffees a day and barely feel a hit, you need to go cold turkey for 12 days. After that the impact of coffee should be more significant. (1) (edit: sources state 2 weeks to 2 months in fact!)
I decided to give it a go a few years ago and fuck me. The come down sucked. The first few days were okish. Slight headache.
But from the 3rd day for about 4-5 days. I had a headache that'd compete with my past migraines. I was nauseated and was throwing up several times a day. These stopped at roughly day 8 or day 9.
I didn't consider myself a heavy coffee drinker either. Perhaps 2-3 coffees a day, these were double shot espresso's.
My colleagues hilariously wrote a medical blog to document my descent into madness. I'll try and dig it up.