First I'll start with this statement :- Dobro is one of those names that have become so synonymous with the product that most people use the brand name rather than the actual name. Dobro was a maker of metal cone resonator instruments, not all metal cone resonator instruments are or were Dobros. it can get a bit more confusing here as to the brand name Dobro only refers to the single cone inverted resonators not any other other sort!!! IT DOESN'T MATTER! It's the brand name for a line of chordophones; not the name of the product! Now onto a tricky story involving quite a few famous brands. (There is a book by Bob Brozman that is suppose to cover all of this but I can't get a copy, so have not read it and am having to piece the story together from other sources that are often conflicting, trying to paint one side or the other as the reasonable party and the other as the difficult one? I'm going with what I have read that makes the most logical sense to me) 

The story starts in the early 1920's when George Beauchamp a California Vaudeville artist got John Dopyera a local luthier to start experimenting with variants of August Stroh's Horn Resonators and Banjo technology to make Guitars that were loud enough to compete with the brass instruments on stage. They had some success and in 1925, formed The National Stringed Instrument Corp. in California. It started out making steel bodied Banjos and Banjoleles, for which Rudy Dopyera obtained a patent, (applied in 1927 and given in 1929 but staying with National), so it wasn't only John doing the inventing. Its first Resonator Guitar was produced in 1927 based on the three cone design patented by Dopyera, which they called the Triolian. Dopyera much preferred this design as he considered it had a much better tone, however prior to coming up with it he had experimented with a larger single cone that, though not as good tonally, was actually louder, (and would be much cheaper and simpler to produce). Beauchamp who had started out looking for volume and could see the greater mass market potential of the single cone took the single cone design that the Dopyera brothers hadn't patented and did so under his own name. He also switched production to the single cone resonator, (though continued to call them the Triolian?) after only a few, (maybe only 12?) three cone guitars were produced. Single cone Ukulele production was also started around this time, with one of these models also being called the Triolian. Upset with this John Dopyera left in 1928 and founded a new firm, Dobro with his brothers Robert, Rudolph (Rudy), Louis, and Emil (Ed). John and the other brothers all had stock in National at the time of leaving, which they kept, but National held on to the existing patents, both the single cone patented by Beauchamp and John's original Tricone. John however had been working on a new design of single cone though, which was inverted and used a "spider" bridge to transmit the string energy and quickly patented this for use in the new Dobro instruments. These new instruments were also wood bodied, (made by Regal), as oppose to the steel bodies ones made by National. Though they made Guitars, Mandolins and Ukuleles, plus John invented the Tenortrope, (a kind of tenor banjo but with a Resonator instead of a skin based drum for a body), Dobro never made the more traditional Banjos

Exactly how it happened I'm not sure, prolonged legal actions? The depression causing a marked drop in sales for the still expensive steel bodied National instruments, (Dobro only had wood bodied instruments)? Better licensing and marketing of the Dobro resonators through big firms like Regal and Montgomery Ward? Better predictions on where the market was going to go? Maybe all of them but in 1934 the Dopyera Brothers regained control of National and it became the National-Dobro Corp. Beauchamp left the company and went off with Adolph Rickenbacher, (who may have been another director of National and was certainly the person behind producing the cones and steel bodies for National, and the cones and spider bridges for Dobro), to found what was to become Rickenbacker Guitars, (one of the early pioneers of electric guitars which was the technology that did for the resonator guitar over the next few years)

It was always the case that a lot of the manufacture was done by others for both companies, certainly all of the metal work and a lot of the woodwork too, so with the shrinking market due to them falling out of favour and competition from electric amplification Regal, who was now doing pretty much all the manufacture, was granted an exclusive licence to the inverted cone technology and the Dobro name in 1937, and from then on until 1941 when WWII stopped all manufacture all Dobro instruments were by Regal; production was never restarted by Regal after the war.

Exactly what happened in the late 30's I don't know and I don't know what happened to John, Emil, Robert or Rudolph, but alongside Regal taking over the Dobro branding Louis Dopera, (with Victor Smith and Al Frost) started a company called Valco making ordinary acoustic Guitars, resonator Guitars, electric lap steel Guitars and amplifiers, under a variety of brand names including Supro, Oahu, and National, (they also made Airline branded equipment for Montgomery Ward), with National being their premium brand name, (solid body electric guitars were added in the 60's but I don't think they ever did Ukuleles?). Valco bought Kay in 1967, however the merged company was not successful and in 1968 was taken over in turn by Weiss...

Emil (Ed) Dopyera started production of Guitars, (but not Ukuleles) again in 1959 under the branding of Dopera's Original and Dobro, (presumably because with Regal gone the name was theirs again?) but he then sold the name to Mosrite Guitars. In 1967 he started up again, this time with his brother Rudolph, (Rudy), with a company called the Original Musical Instrument Company (OMI) to manufacture resonator Guitars again using the brand name Hound Dog (as Mosrite had the Dobro name). In 1970 with Mosrite going into liquidation, they bought back the Dobro name. In 1993 Gibson bought OMI and with it the Dobro name and Gibson still make inverted cone Guitars, (but still not Ukuleles), branded Dobro 

The current National Reso-Phonic Guitar Co. though it likes to allude to the history and makes copies of the original National Resonators, (including the Ukuleles), doesn't have any real link beyond geography