Short answer
In General American English, a /t/ will tend to be voiced when it occurs after a nasal and before a weak vowel. This sounds a bit like a /d/ to many listeners. If you're looking at an IPA transcription in a good dictionary, such as Cambridge Dictionaries Online you should see that the /t/ has a little diacritic underneath. The symbol will look like this: t̬. Voiced /t/s have no aspiration as is explained further below.
Detailed answer
It is a well known fact that /t/ is liable to be a voiced tap when it occurs after a vowel and before another weak vowel. So for example the /t/ in water will be pronounced [ɾ] in Gen Am. The Cambridge Dictionaries Online transcription for this is /ˈwɑː.t̬ɚ/. That little downwards arrow there, which should be directly underneath the 't', shows that this /t/ is likely to be voiced.
However, what's less well-known is that /t/ is also likely to be realised as voiced when it occurs between certain voiced consonants and a weak vowel. So for example /t/ may also be voiced when it occurs after /r, l, m, n, ŋ/ and before a weak vowel. So we will probably hear a tap or other voiced allophone in the following words:
If you look at the transcriptions for those words in the links you will see the voiced-t symbol, /t̬/. You can also hear a tap in most of the American English pronunciations there too.
OK, but why is there no aspiration there?
The English plosives /p, k, t/ usually occur with some aspiration before vowels (when articulated as plosives, not taps or glottal stops). This may be light or it may be strong depending on it's position in the word. These plosives are usually strongly aspirated when occurring word initially, at the beginning of a stressed syllable or when occurring before a strong vowel. Elsewhere, when occurring before a vowel, /t/ will be lightly aspirated. The exception to this is when the /t/ is preceded by an /s/. When this occurs the /t/ will be completely unaspirated, and will sound like a /d/ to many listeners.
Now this is what actually happens when we get an aspirated consonant in English. The plosives /p, k, t/ are typically voiceless. We make them by forming a complete blockage in the vocal tract which prevents the air from leaving the mouth. Behind this blockage there is a huge build up of air pressure as air is forced up from the lungs. This is called the hold phase of the consonant and with voiceless consonants it is accompanied by silence. This is because there is no air moving out of the mouth and the vocal folds (or vocal chords) are not vibrating. When we finally release the air we get an effect called plosion, caused by the air exploding out of the mouth.
Now usually these consonants occur before a voiced sound, most often a vowel. Vowels of course are voiced and this is what gives them pitch. So when we make a vowel we vibrate our vocal folds. However our vocal folds are generally a bit lazy when we speak English. They tend to kick in quite late and they often finish early too at the end of a word or syllable. When we get a vowel after a voiceless plosive in English there is a small gap before the vocal folds actually start vibrating. This lag is called the voice onset time, or VOT, for short. So the VOT is basically a gap between the release of the plosive and the beginning of the voicing of the vowel.
Now when we release a /p, t/ or /k/ the air that we release is being forced out of the mouth by air coming from the lungs. During the VOT period, before the vocal folds start vibrating, we can hear the turbulence of the air as it leaves the mouth, because there is no vocal fold vibration to drown the noise out. This turbulence that we hear has an /h/-like quality, and it is this sound that we recognise as aspiration. So basically aspiration is just the air rushing out of the mouth before the vocal folds get to work.
You will have noticed that I said when they are articulated as plosives further above. When /t/ occurs intervocalically (between two vowels) or between /r, l, m, n, ŋ/ and a weak vowel, we see assimilation. This means that the /t/ takes on features from the sounds next to it. In particular, it becomes voiced. In practice this means that we don't stop vibrating the vocal folds for the /t/. They vibrate all the way through from the previous sound, right through the /t/ and continue vibrating for the following vowel. It is this aspect of the /t/ that makes it sound like a /d/ to many speakers in words like water. The upshot of this is that because there is no gap in the voicing between the consonant and the following vowel, there is no aspiration.
Conclusion
In short the Original Poster is completely correct. It is not that they can't hear the aspiration because the /t/ is only lightly aspirated in the words ninety or entertain. It's that there actually is no aspiration in the pronunciations given in the audios from the dictionary. They have used a voiced allophone, instead of [t] for these words. This is not a case of interference from the Original Poster's first language. It's a case of razor sharp ears!
It might be worth mentioning that in British English the use of [ɾ] for /t/ is much rarer. We tend to stick with [t] when American speakers would use [ɾ]. If you listen carefully to the [t]s in the British English versions of the words in the audio, you will be able to hear the light aspiration.
Addendum for phonetics junkies
It might be worth mentioning that when we get at /nt/ sequence followed by a weak vowel, in rapid informal speech, the whole /nt/ sequence might get replaced by a single nasalised tap. So for example, we might see [ˈtwɛ̃ɾ̃i] for the word twenty. This instance of coalescent assimilation will often be perceived as /t/-elision by listeners.
