Yes, and even more to it: set comes with different pieces. What i mean is that the same set may require different items for it. For example sorcerer set: 1 i have is chest, helm and neck, but i also have seen same set with chest ring and helm.
So not only you have more options then 7 armor pieces, as sets may include jewelery too(100% sure, as said, have sorc set on my main for healing), but you can also, if a bit lucky, find set you desire, which will use pieces of equipment not used for other set.
There are 5 piece bonuses too, but yes you can have multiple bonuses. However, many of the bonuses are more situational than flat out dps increases. (Health increase on stealth, knockback if you're under 30 percent health, etc) so they're not necessarily must haves.
Holgrum put some pants on. Holgrum heal me I'm dying. Nag... Nag... Nag...
ESO talked about this in a Q&A, multiple bonuses etc, but later game you will have set bonuses that may require every piece, but would have a more unique set bonus.
Basically we all have access to either 11 or 12 slots that can towards a set. For instance I currently am wearing a warlock 3 piece set consisting of a ring, an amulet, and a head piece. That leaves 9 slots left I can use towards other sets if I want (2nd ring, 1h sword, shield, shoulders, gloves, belt, chest, legs, boots).
If I really wanted to go crazy, I could potentially have 4 different 3 set bonuses as long as I went S/B or dual wield. Do remember though at end game there's 4 item set bonuses and 5 item set bonuses as well. Take a look at the thread I linked and you can see the HUGE amount of choices we all have.
@lunarsyed this thread is from when the game first launched, it was basically a different game then. To answer your question you cannot wear the same gear set twice and get double the effects. Do not wear 10 pieces of automaton.
#1 tip (Re)check your graphics settings periodically - especially resolution.
You can't stack the same bonuses. If you wear 2 of the same gear set, it will only register as 5 pieces. If you get, say, Major Brutality from a skill and from a potion, they won't stack. If you get Major Sorcery from a skill and you wear the Rattlecage set, the Major Sorcery won't stack. However, Major and Minor buffs of something stack.