Between october 7th 1840 and december 12th 1843 The Netherlands had HM the King (Willem II) and HM King Willem Frederik, count of Nassau (the abdicated king Willem I).
In Belgium we've had HM the King (Boudewijn) and HM King Leopold III and since the accession of the current king HM the King (Philippe) and HM King Albert II.
All these examples show that the title of King does not automatically mean that the individual styled as such is also the reigning monarch as is often wrongly assumed.
The Bourbon dynasty had several reigning monarchs of different branches in the 18th century:
The King of France and Navarre
The King of Spain
The King of Napels and or the two Sicilies
The Duke of Parma and Piacenza
The Habsburg dynasty in the 19th century also had multiple monarchs:
the Emperor of Austria (King of Hungary)
the Emperor of Mexico
the Grand Duke of Tuscany
the Duke of Modena
Had Catherine the Great had her way the Romanov dynasty would have had two imperial branches:
Tsar/emperor of Russia (Alexander I)
Emperor of Byzantium (grand duke Constantine)
For a short while the house of Hesse had several reigning branches:
King-elect of Finland
Grand duke of Hesse und by Rhein
Landgrave of Hesse(n-Kassel)
im not including the various junior branches like Hesse-Phillippstal and Hesse-Phillippstal-Barchfeld.
It has struck me that between 1848 and 1867, the house of Wittelsbach actually had TRHEE men styled with the title of King with the qualification of Majesty. From the abdication in 1848 of Ludwig I to the death in 1864 of Maximilian II, one had HM King Ludwig I of Bavaria, HM The King of Bavaria (i.e. Maximilian II), and HM King Otto of Greece (until deposed in 1862, he was styled as HM The King of the Hellenes).
Then, between the death in 1864 of Maximilian II to the death in 1867 of Otto, one had HM King Ludwig I of Bavaria, HM The King of Bavaria (i.e. Ludwig II), and HM King Otto of Greece.
Have there been other periods in royal history where one had more than two men of the same house styled with the title of King? The only other example I can think of would be in the family of King Christian IX of Denmark (the so-called "Father-in-law of Europe"). In 1863, his second son, Prince William, got elected to start a new Greek royal dynasty, after the afore-mentioned House of Wittelsbach got deposed. He became King George I of Greece on the occasion.
Later on, the house of Glücksburg branched further out, when in 1905 Norway ended her personal union with Sweden and invited Prince Carl of Denmark (second son of the future King Frederik VIII) to become her first king. He thus became King Haakon VII.
So between 1905 and 1906, one had HM King Christian IX of Denmark, HM King George I of the Hellenes, and HM King Haakon VII of Norway sitting on royal thrones.
The house of Coburg might also count, since Prince Ferdinand of Saxe-Coburg and Gotha (Kohary) received the title of King Consort of Portugal after marrying Queen Maria II and fathering sons. And of course the said sons and grandsons through them were kings of the house of Coburg (as well as Braganza). Since this house also reigned in Belgium and Bulgaria, this might also count. The trouble is that Bulgaria was originally a principality, and got elevated to a kingdom only in 1908. But as of then, the Coburgs were still reigning in Portugal (although that situation would soon change), as in Belgium.
Are there other examples? I don't care whether the men in question were enthroned or not: even abdicated kings continue to be styled with their exalted titles (as in presently the case in both Spain and Belgium).