Obviously the reigns of the two English kings you mentioned could not have overlapped. That being said, given the strategic marriages made by the other children of King Henry II of England and his queen consort, Eleanor of Aquitaine, the likelihood of the reigns of two or more sovereign children overlapping would have been great.
What about their half-sisters, the daughters born to their mother's first marriage (to King Louis VII of France)? Didn't they make strategic marriages as well? Did their reigns overlap with any of their half-siblings?
As for the Iberian peninsula: the situation would have been similar. When three or more of your children sit on thrones, there is a high likelihood of two or more simultaneously being reigning monarchs, at some period or another.
Those were the days ...
Richard Lionheart I of England and John I of England:
their brother Geoffrey was Duke of Brittany and their sisters:
Mathilde duchess of Saxony and Bavaria
Eleonor queen-consort of Castille
Joan, queen-consort of Sicily
Henry I of Castile
sister and successor Berengeria Queen of Castille
sister Urraca Queen-consort of Portugal
younger sister Blanche became Queen-consort of France after Henry and Urraca died and Berengeria abdicated. But another sister Eleonor was Queen-Consort of Aragon.
Blanche's oldest son Louis IX was king of France and his youngest brother Charles I was king of Naples.
Children of Juana and Philip the fair:
Eleonor queen of Portugal 1518-1521 /queen of France 1530-1547
Charles King of Castille and Aragon 1516-1556/Holy Roman Emperor 1519/1556
Isabella queen of Denmark 1515-1523
Ferdinand king of Bohemia, Hungary and Croatia 1526-1564 and Holy Roman Emperor 1556-1564
Mary Queen of Hungary 1515-1526
Catherine queen of Portugal 1525-1557
They are probably one of the very few couples who had all their children end up on thrones.
Margaret Tudor, queen of Scotland 1503-1513
Henry VIII king of England 1509-1547
Mary Tudor, queen of France 1514-1515
Frederick William III King of Prussia 1797-1840
Wilhelmina, Queen of the Netherlands, Grand Duchess of Luxembourg 1814-1837
Augusta Electress of Hesse 1821-1841
Frederick William IV king of Prussia 1840-1861
William I King of Prussia 1861-1888, Emperor of Germany 1871-1888
Charlotte/Alexandra Feodorovna, Empress of Russia 1825-1855
Alexandrine, Grand Duchess of Mecklenburg-Schwerin 1837-1842
Frederick King of Sweden 1720-1751/ landgrave of Hesse 1730-1751
Sophie Charlotte, duchess of Mecklenburg-Schwerin 1704-1713
William VIII Landgrave of Hesse 1751-1760
Marie-Louise, Princess of Orange, Princess of Nassau-Dietz 1709-1711 regent 1711-1731 and 1759-1765
The discussion on branching out reminds me: what are the periods in European royal history where you had two or more siblings in the same family simultaneously sitting on thrones? It was a common occurrence in the past -- especially before World War I. But of course, this has become an increasingly rare occurrence in modern history. It's easier to go back in time, through the decades, but I've come up with the following (queens consorts are acceptable) --
1972-1973: Anne-Marie was the exiled queen of Greece, during the first year of her eldest sister's reign as Margrethe II of Denmark, the country's first queen regnant in 600 years. The Greek monarchy was abolished in 1973, two years before Juan Carlos was called to the Spanish throne as king, following the death of Generalissimo Francisco Franco. It was then that King Constantine II's sister Sofia became queen. But that example doesn't count, since their reigns never overlapped.
1964-1993: Baudouin was the king of the Belgians while his sister Josephine-Charlotte was the grand duchess consort of Luxembourg
1993-2000: Josephine-Charlotte was the grand duchess consort of Luxembourg, while her brother Albert II was the king of the Belgians
1913-1918: Kaiser Wilhelm II sat on the German imperial throne, while his sister Sophie was the queen consort of Greece
1881-1894: Dagmar was the empress consort of Russia (Maria Feodorovan), while her brother George I was the king of Greece
1906-1910: George I was the king of Greece while his older brother Frederik VIII was the king of Denmark; also, their sister Alexandra was the queen consort of Great Britain
1910-1936: George V was the king of Great Britain, while his sister Maud was the queen consort of Norway
1912-1947: Christian X was the king of Denmark while his brother Haakon VII was the King of Norway
1894-1917: Ernst was the reigning grand duke of Hesse-Darmstadt while his sister Alix was the empress consort of Russia
1848-1862: Maximilian II was the king of Bavaria, while his brother Otto was the king of Greece
1859-1861: Elisabeth was the empress of Austria-Hungary, while her sister Marie Sophie was the queen of the Two Sicilies
Can anybody fill me in on additional examples within the last 200 years? I'm less familiar with the smaller German houses.