It depends what the converter is doing. If it is literally just converting then some TVs don't accept 480p, 480i, 240p over HDMI.
You need a VGA to HDMI upscaler (or an HDMI to HDMI upscaler that accepts lower resolutions). That way the upscaler will output a common 720, 768 or 1080p signal your TV will understand.
Getting a new TV is unlikely to solve your problem, as most modern sets don't work with low resolution HDMI.