Can I overload my tank with corals?

I agree with everything said so far. That said, adding 3-5 corals at once is not an issue IMO, especially if they are softies like mushrooms/zoos etc. Although maybe there are other considerations to take into account with softies, they're not my particular cup of tea beyond a few zoos.
My one caution is to be careful adding some of these though because they can grow like weeds and easily take over a large portion of the tank. Especially the GSP and mushrooms. If you do some searches you will find plenty of stories of people who have had tanks on the way to being completely overrun by these corals.

On the other hand I wouldn't recommend adding a large number of stony corals at once either due to the previously mentioned impact on calcium/alkalinity levels. It can actually be quite tricky to handle a large supplementation scheme since you have to balance pH, and not have large swings especially with alkalinity over the course of the day among other little things that crop up.

Some people can manage dosing every single day 2-3 times a day (without fail) to prevent swings in alk etc but believe me, I've been fighting chemistry in my tank for a long time and I feel like I have a pretty good grasp of the chemistry. It's not always easy and not every tank is the same so there's no hard and fast rules to follow.
There are lots of details that are easy to overlook or underestimate. I wish I hadn't added so many SPS to my tank so fast (over the course of a couple months actually which I thought was "slow" but apparently not for manual dosing). I really needed to have an automatic doser or calcium reactor setup to manage the load I put on my tank.

Live and learn, and hopefully pass on your mistakes to others before they make the same ones.

 

members of the same family cannot sting each other. some softies cant do much to anybody.