first of all let me mention that the recent rise was not a rise at all but in fact a fall. in other words value of Tether dropped so in order for altcoins to keep their value up they needed to rise against Tether and that is what happened here.
but lets get to the middle part of OP comment. when bitcoin rises (assuming Tether is worth $1) altcoins seemingly rise up but not because they keep their value, in fact they fall in value against USD and BTC but because this fall is not equal to bitcoin price rise, the final result is a rise. for example if bitcoin is $1000 and some altcoin is worth 0.1BTC and then bitcoin goes up to $2000 that altcoin, if it remains 0.1BTC will rise to $200 which is impossible for that coin to happen because there is nobody buying it. what happens is that rise of bitcoin from $1000 to $1000+ will start creating arbitrage opportunity in ALT/BTC and ALT/USD and as traders arbitrage this the final value of that coin will go somewhere between $100 and $200 and closer to lower bound. so the final result will be an altcoin worth around 0.06BTC which is a big fall but thanks to arbitrage traders the value of it rose a little to $120 in a fake manner.
Btw tether didn't fall so much to cause almost 1/2K rise in bitcoin's price, tether got -1% but bitcoin got more than 10%.