Many speakers of Gen Am and also speakers of British Englishes, including some young RP speakers, use a hard attack on the second word to separate a word-final and word initial vowel. For a minority of speakers this also occurs after the definite article. A ʜᴀʀᴅ ᴀᴛᴛᴀᴄᴋ is when a speaker uses a glottal stop, [ ʔ ], at the beginning of a word starting with a vowel*. So, for example, instead of saying [ɛnd] for the word end, a speaker using a hard attack would say [ʔɛnd]. In this syllable initial position, the glottal stop will not be recognised as a /t/.
Speakers who use a hard attack to separate the vowel in the from a following vowel will therefore say:
This is a relatively new phenomenon, at least in RP. Will pedants decry any new and novel development in the language? Of course they will!
But are these speakers making mistakes? No! The rules of language are the rules which describe what real speakers of the language do. The Original Poster has therefore observed a real rule underlying such speakers' speech.
* Note that this is a hard attack, not a heart attack!