Says it all in the title, were there ever Medieval Kings who were bastard children who through solid governance, stellar reputation or skill on the battlefield succeeded as monarchs despite their humble background? If so, how common was it for said monarchs to have all three qualities (like Snow in GOT)?
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John I of Portugal.
He was the bastard son of Peter I of Portugal, that was succeeded by his legitimate son Ferdinand I (John's half brother) in 1367. When Ferdinand I died in 1383, he left only a daughter that was betrothed to the king of Castille.
The king of Castille proclaimed himself king of Portugal and invaded the country. Getting in charge of the defense of the country, John (at the time Grand-master of the military order of Avis) successfully repealed the invading armies with the help of english troops.
In 1385, John was acclaimed as king in a conclave and starting the Avis dinasty. In 1386, the treaty of Windsor is signed between England and Portugal, founding one of the oldest active military alliances.
https://en.wikipedia.org/wiki/John_I_of_Portugal
https://en.wikipedia.org/wiki/1383%E2%80%931385_Portuguese_interregnum
(btw being a bastard of a king is hardly a humble background)