Clearly, #"tert-butanol"# has some similarity with the water molecule. And that similarity is the hydroxyl group. You should look up the aqueous solubility. It would not be immiscible; by the same reasoning it would not be infinitely miscible, as is methanol and ethanol.
A good comparison would be the solubility of #"n-butanol"# and #"sec-butanol"# in aqueous solution. As a prediction, which certainly might be mistaken, #"tert-butanol"#, might be more soluble than #"n-butanol"#, but as chemists, as physical scientists, we should always seek out the data to inform our arguments, and never the other way round.