In India, I have met a few people who speak with an American accent. I have noticed that not all the words they pronounce have this foreign accent. An Indian accent on a word slips in, here and there. And then, their ‘American’ accent sounds artificial. 😦
Does one have to speak like a native speaker? The short answer is: no. There are many kinds of English out there. We have Japanese English, Spanish English, and of course, Indian English. All these different kinds of accents are fine. Your accent reflects the place and culture you belong to. In the US, there are many accents. All these accents are fine. And so is yours.
What is important is that your pronunciation should be understandable. To better pronounce a word, you can focus on syllable stress and the phonemic chart. Do check out the other blogs on this page relating to pronunciation.
Here is a charming song on the differences in how we pronounce words: