I'm listening to a native French speaker reading the following two sentences:
Ma mère est anglaise
Et tu parles français avec ton père et anglais avec ta mère ?
In the first one, not surprisingly, "t" at the end of "est" is pronounced because it's followed by a vowel. But the speaker in the second sentence does not pronounc "t" in "et anglais", even though it's followed by a vowel. Why?
The liaison rules are not that simple (same page in English.)
There is never a liaison after et while the liaison is possible, but not mandatory, after est.
A final -t often has tricky liaison rules, e.g.:
Here's another explanation of liaisons which I think is a bit more accessible, and also includes sound files.
To summarize, there are three types of liaisons: required, optional, and forbidden. The t in est anglaise is an optional liaison, while the t in et anglais is a forbidden liaison.
Maybe it will help if you understand where the liaison rules come from (they can seem a little arbitrary).
As in English, the reason most silent letters occur is that these letters were still being pronounced back when the spelling system was being developed. In French, the spelling system was developed before the loss of final consonants, and this is why there are so many words whose spelling ends in a final consonant which is not pronounced.
As the consonants gradually began to become silent in word-final position, they were protected from this change when they were followed by a vowel in the next word. This is because then, as now, French speakers constantly resyllabified words, so that e.g. il était une fois was pronounced i-lé-tè-tu-neu-fois, and thus the t wasn't "word-final," and so was protected from the erosion of final consonants.
(Something similar happens with the r sound in British English: no r is pronounced in mother, but a "linking r" is pronounced in mother of. Another similar thing happened hundreds of years ago in English: the English indefinite article a/an used to be "an" in all contexts [ignoring some complications], but the "n" gradually eroded away, except before vowels; in terms of spelling, we treat this differently from the French, writing the word as "an" in front of a vowel and as "a" otherwise, but in terms of bpronunciation, it's the same phenomenon.)
In short: liaison sounds are sounds that were lost in the distant past, but that were preserved in front of vowels. Since the writing system was developed before the sounds were lost, it helps us figure out when to make a liaison (this is presumably why no one has ever gotten around to updating those silent letters out of the spelling).
The phrases that still admit mandatory liaison today are the phrases that constantly come together in groups, e.g. des_amis, les_armoires, vous_avez. Roughly, the more frequent a "grouping" is, the more likely liaison will occur. In the casual language, the liaisons one hears most frequently are these obligatory liaisons, and liaisons in fixed expressions (de temps_en temps). I find the rule "how likely is someone to hear this fixed expression with this fixed meaning?" to be a reliable indicator of how often a liaison occurs.
So back to your question -- why doesn't all the same apply to the word "et"? Above, I mentioned that the reason most silent letters occur is that they were still being pronounced back when the spelling system was being developed. The "t" in "et" is not like that; the word for "and" had been pronounced /e/ for hundreds of years before the spelling system developed, in all contexts, so by the time the sound changes mentioned above took place, there was no sound there left to save, and hence there is no liaison today. The silent "t" in "et" is due to early French grammarians, who romanticized the Latin language, in which the word for "and" is "et"; they decided to throw an extra silent "t" in the spelling of the French word which is unrelated to the "t" in e.g. "était" which really was still being pronounced at the time. (English has quite a few indefensible spellings for the same reason -- for instance, the silent b in debt was reinserted to pay hommage to the Latin etymology of the English word).
