Sort of.
Octavian was Caesar's closest male relative just as Hadrian was Trajan's closest male relative. Adoption from within the same family was more common than from without. Adoption of the 'best' was simply a case of expediency.
Hadrian had no living male relative so to aviod a civil war he adopted Aelius Verus who then died and the adoptions by Aelius' replacement, Pius, were of Pius' and Verus' heirs. In fact Hadrian was the only emperor up to then who adopted someone from outside his family.
Adoption was usually, but not exclusively, really just a method of making sure the inheritance stayed within your kinship group.
There are questions as about Hadrian ever being adopted by Trajan or not. There is a chance he was actually became emperor because of the support of the army.
I have thought a bit, and I think I can come up with a simple timeline for a successor to Aelius' who is someone different. The main change is in 175 the reports of Aelius' death do not come. Thus Avidus Cassius is not crowned Emperor by his own troops and you don't have him killed when he finds out Aelius is still alive.
175 AD-176AD while on campaign against the Marcomannic People, Marcus Aelius becomes very ill, causing questions to arise about the successor of the Emperor. Faustina is very worried about her husband, also believing Cassius should become the protector while her son Commudus was still young. Sends a secret letter to her husband telling him of her ideas. Later in the year Marcus Aelius is able to defeat the remaining tribes in the region. Bring in the area of modern Slovakia and Czech into the Roman fold.
177AD- 180AD Marcus Aelius returns to Rome to celebrate a Triumph of his victory in Rome. Aelius slowly realizes his own son is not fit to rule at that time. He meets with Cassius and makes a deal with him. Cassius will succeed him, but he must name Commudus as his heir. Cassius agrees to the terms.
180AD- Marcus Aelius passes away, and is replaced by Cassius who names Commudus as his heir. A small group tries to revolt against this action, lead by Faustina who worries that her son will never become emperor. The revolt is quickly put down.
180-195- Cassius rules rome, in a wise fashion avoiding the pitfalls of Commudus. He pulls back to create a more defensible position against Parthia and improves Hardians Wall. At the same time, he campaigns against the German Tribes with positive success, bring the area of modern Austria in to German fold, after putting down one last rebellion of the Macromannic tribes. Commudus dies in an accident in 192, causing Cassius to have to decided on an heir. Cassius in time decides to take Clodius Albinus as his heir. Cassius dies of natural causes in 195.
195-222- Clodius Albinus rules Rome, and to strenghen his positions, he makes a deal with Septimius Severus. The deal in the long run makes Albinus adopt Alexander Severus along with his Adoption of Gordian II. Clodius works with Septimius forces to defeat the remaining Germanic Tribes, bring all of Germania into the Emperor.
222- 258 - Alexander Severus and Gordian work as Co-Emperors, with Alexander. The two work well together, despite some slight disagreements. Gordian manages to put down a revolt in Germany, and also manages to strengthen the defenses in the East. While at the same time, Alexander's domestic policies improve the economy of Rome.
258- Gordian dies, and Alexander knowing the army wants a military man as co-emperor selects Diocletian a rising star in the army to be his co-emperor and heir.