As a tank, I would say your usually in the clear to lay down a regrowth early against a boss. Any good tank will front load alot of threat right away so the healer can put down as much healing is needed without pulling threat. Against multiple mobs it would probably be better to be a little cautious initially since warrior and druid tanks cannot frontload alot of threat on all targets right away. How careful you need to be depends on the gear and skill of the tank.
I don't know enough about multi mob tanking with warriors to give a good evaluation of their threat levels on multiple mobs at the beginning of a pull, but as a druid if I'm only tanking 3 mobs, the healer is usually in the clear 2 seconds from contact (when the first mob is within melee range of me). Every mob above 3 you should add 1.5 seconds in my experience. It can vary of course depending on the type of mobs being tanked (melee/ranged) but it's a good rule of thumb to start with since it will give the druid time to land a mangle on the primary target, a swipe on 3 targets and a global dooldown to switch targets and swipe again if more then 3 mobs to build some initial threat on all mobs being tanked.
Given that every point of healing is the same as 0.5 points of damage in terms of threat, and tanks/healers usually have threat modifiers on top of that, front loaded threat on a boss gives more then enough room for the healer to do what is required. For example, if my first mangle crits the healer is clear to do over 8000 healing in the first global cooldown before being at risk of pulling threat. Against 3 mobs, swipe spam alone will usually sustain 400-500 tps on all mobs, clearing the healer to sustain 1300 hps from 2 seconds after contact. More mobs means threat needs to be more spread out, so both more cooldowns are required to get inital threat on all targets, but also the tps on the primary target is reduced, since the tank is spending more time to boost threat on the other mobs, so the dps needs to be careful.
Remember that my numbers are from me personally though and not all tanks will be able to sustain or frontload the same amount of threat. Make a habit of inspecting the tank before each group and try to evaluate what they are capable of given their gear compared to previous tanks you have healed. Generally though, you shouldn't worry about threat on bosses since tanks can usually front load enough threat to give you alot of room to work with.
If your tank isn't skilled enough to know how to maximise threat genereation though, then you'll have issues regardless. A tank needs to be able to maximise the window the healer has from when they can start healing to when the tank dies from not being healed. If your tank is making that a very small window, then the group is probably doomed.