I am an Indian citizen and have a Hungarian residence permit valid till the end of July. My Swedish residence permit kicks in on the 15th of August. Its too expensive to travel back to India and was wondering if I can stay on for these 15 days in the middle, without a visa. Please let me know if anyone has done this before! Thanks!
Not without an additional visa. I guess however, since it is just to bridge the gap between two residence permits, that it should be no problem for you to get a short-term (type C) visa for the period from August 1st to 14th.
Never overstay your permission to stay anywhere in any circumstances.
When you apply for a visa, your history is looked into in detail, and they are looking for reasons to reject the application. If they find that you have ever previously overstayed your permission to stay anywhere in the world, you just gave them a reason to reject your application.