Learning the rules of phonology in a language might make the lowest levels of acquisition easier in some languages that have complicated phonologies (such as first-year Arabic or Klamath), perhaps 10% easier. It could help to reduce the chaotic appearance of the inflectional system, all of that complicated stuff about vowel changes and glides coming and going (in Arabic). You will still not sound like a native speaker even a little bit. For English, no amount of studying the putative phonology behind word pairs like "cone, conic", "obscene, obscenity" will make you sound like a native speaker. I have (or had) a reasonably good understanding of Klamath phonology, and not the slightest idea what the language actually sounded like.
You might also intellectually master knowledge of the acoustic and articulatory phonetics of a language, that is, read all of the phonetic literature on the language – read myriad articles that measure amplitude, pitch, formants, duration in some language. That will still not tell you what the language actually sounds like. The only thing that will help you sound more like a native speaker of the language is being exposed to native speakers of the language: you have to speak the language, with native speakers. You don't have to live with / among them, but you have to intensively interact with them.
India presents the additional complication that the national standard for pronunciation isn't the same as that of the US, UK, Australia, that is, Indian English is a separate language. I don't know of any research into the phonetic and phonological properties of Indian English monolinguals or at least first-language speakers. I would take differences in in-class vs. out-of-class performance to be evidence that the person has not natively mastered English phonetics – unless the instructor is actually trying to teach American English pronunciation, and is a fluent Indian English speaker.