Maybe you can guess some hand-picked examples, but that's it. Your analysis of 病 is incorrect. It's a phono-semantic compound of 疒 (sickness) and 丙 (pronounced "bing"). From this you can deduce that 病 is pronounced similar to "bing" (or used to, in ancient Chinese) and has sickness-related meaning.

Some characters actually do combine the meanings of several radicals, for example 孝 is a child carrying an elder on their back or something; it means "filial piety" (respect for elders). But there are not that many of them, and you have to know the final meaning anyway, otherwise it's hard to guess.