Yes, it almost always happens to me too, in sentences that begin with "Hundene". It's a fault with the Text-To-Speech system. The correct pronounciation is indeed just "Hundene" and not "Tundene". I've become used to it, and now whenever it says "Tundene", I just hear and think "Hundene" in my head.
You could say "Jeg var nær ved at blive ramt af en bil." which means "I was nearly hit by a car." But in normal spoken danish most people would not say "nær ved" in that situation, but "Jeg var lige ved at blive ramt af en bil." - The meaning is the same, but it's an "older" way of saying it.