They do tend to run together a bit, but the t sound is still there the way I pronounce it. However, it is possible that it might be barely there or not there at all. Example:
He said he would see me the nex' day.In that case there would be no pause between the words.
~R
What you say about assimilation is true in much current speech; and many speakers aren't aware it's happening. To take another example, it's quite hard to convince a native speaker that when they say 'fine' in an utterance-final position the noise they make is /faɪm/ ('assimilation' not to a following phoneme, but to a following mouth position - closed). But just because assimilation can happen, and in some cases is likely to happen, it doesn't always happen. In careful speech it may not.
b
There's a great deal of room for individual differences in pronunciation, of course. Certainly, I don't pronounce the "t" in "next day"; and exactly how I pronounce the final phoneme in "fine" depends on to what extent I am emphasising the word.