Control and removal are your biggest counters. Slivers have specific cards against them, of course, but very limited.
While spot removal can be annoying, all your slivers are useful creatures. It's not like some decks that have a keystone creature you need. Here, every sliver counts, and if they kill one, you can replace it, but losing one of your +1/+1 slivers at a key moment can turn the tide in favor of your opponent. You should always be mindful of which cards their deck has and if they have cards in their hand and mana available.
Hard control, such as the classic prison enchantments/creatures, counterspells, bouncing, and such, are much more annoying. While in Commander, you have Crystalline Sliver and Ward Sliver, they are not legal in Modern. Right now, there are no slivers that give themselves hexproof.
--Cavern of Souls will be your main card against counterspells. After all, you're playing a tribal deck, and everything is sliver. It's a land, and it gives you mana. It's a turn 0 drop, impossible to counter itself. You drop that on your first turn, and control decks lose a lot.
--Rhythm of the Wild is another way to prevent countering with the added bonus of haste or +1/+1. Only Destiny Spinner is cheaper, but Rhythm of the Wild is much better for a sliver tribal deck. The other alternative is Leyline of Lifeforce if you happen to draw it on your starting hand.
--Aether Vial is your second-best card against counterspells. You do not cast the creature, you put it into your battlefield. This is different from Collected Company and Realmwalker, which allow you to play slivers from your library, but you are technically casting them. At the same time, Aether Vial won't trigger Lifecrafter's Bestiary. On the plus side, you can drop slivers during your opponent's turn.
--Ascetism is the only viable way to have hexproof on your slivers, and it's expensive mana-wise. A variant is Privileged Position, which affects all your permanents. It's not very viable. There's nothing for less than 5 mana that gives permanent hexproof with one single exception...
--Lost in the Maze is the only card under 5 mana that gives permanent hexproof, but the caveat is that it only works on tapped creatures. If you have a mana sliver, you can always tap any sliver, making it very viable for 2 mana, plus extra effects if you want.
--Heroic Intervention remains, so far, the best option to save your bacon in most situations, as all your permanents gain hexproof and indestructible. They can still be hit by wide-area spells such as Languish or Evacuation.
--Opaline Sliver: Like most of your anti-control slivers, this one only makes it more annoying to get rid of your creatures. In this case, you draw a card.
--Diffusion Sliver and Unsettled Mariner work in a very similar manner. In essence, Diffusion Sliver gives your slivers Ward 2, while Unsettled Mariner gives your permanents (that means lands, artifacts, etc) Ward 1.
--Kira, Great Glass-Spinner is not a sliver, but it adds an effect to all your creatures. The biggest issue is that Kira, Great Glass-Spinner is vulnerable to common mass damage such as Pyroclasm and Kozilek's Return, which are staples for Eldrazi and red decks in general.
--Sedge Sliver it's no Crypt Sliver but if you have a swamp it's a lord (+1/+1) and allows you to regenerate. I find it very situational, and the 3 mana cost is not the best.
--Darkheart Sliver again, this simply punishes your opponents for trying to get rid of your slivers. The sacrifice ability is unavoidable, which means that you can always respond to anything your opponent does with it. It's situational.
--Frenetic Sliver: You MUST include this one in your mainboard against heavy control, heavy removal, and mass removal. It makes removal useless once you learn how to flip coins. Coin flipping, no matter what people say, is not random. It's based on pure physics. If you learn to apply the same force all the time when you flip, you can calculate how to make it land heads or tails, but it requires practice, oh, and it's perfectly legal :)
Get yourself a custom coin, preferably big and heavy (easier to predict the result), and practice. There is nothing in the rules that says how to flip a coin; the only rule is that the affected player flips the coin and must call head or tails while it's in the air unless the card specifies otherwise. Since coin flipping is an unusual and old mechanic, there are only 3 rules affecting it, none of which specify how to flip, so you can take advantage of this loophole. And yes, I am aware that the "cheat flip" exists (snapping your fingers makes the coin tumble, but not flip as it goes up, so it always faces the same direction), but that's not flipping, and you may get called out.
Now it's time for you to defend yourself against control.
--Leyline of Sanctity, Teyo, the Shieldmage, Witchbane Orb, Ivory Mask, Orbs of Warding and Crystal Barricade are the most viable solutions. Of course, it depends on what the enemy is running. Unfortunately, monored burn decks now have Wear / Tear on their sideboards and run dual lands precisely to get rid of your hexproof.
--Nephalia Academy: Is Thoughtseize a pain in the ass? We thought so too. Put an end to that thought-stealing nonsense once and for all. There's a reason this card has been a staple since it got printed, and why every deck can, runs it. It's busted.
--Veil of Summer and Surge of Salvation remain the best way to gain hexproof with some extra benefits even if it's just one turn. Veil of Summer is a popular sideboard staple.