Yes, it is certainly possible to convert an existing non-responsive theme into a responsive one; however, it’s not always worth it. The amount of effort needed for conversion will depend on many factors such as the current structure of the theme, the amount of non-standard pages, the desired output of UI elements on mobile devices, etc.
While retrofitting a theme with responsive capabilities might look like a quick and cheap option, usually creating a new responsive theme from scratch using a responsive framework proves to be much more effective in the long run. Full-featured responsive frameworks provide a wide array of well-documented and tested UI components such as grids, buttons, tables, etc., that will work in different scenarios on a large variety of screens and devices. This not only allows for more flexibility and easier customization when designing for both mobile and desktops, but also delivers a better user experience.