So I'm clear I understand the question: are you asking if there have ever been a pair of non-successive heridatary rulers?

If so, the answer is yes.

There have been many, many, resurgent dynasties throughout world history. Chinese history has some amazing rise-and-fall-and-rise-again back and forths between warring families, and European monarchies were often traded back and forth between warring families or branches of the same family. It's also hard to define 'dynasty' - many rulers have claimed rather dubious inheritance rights and are hard to consider part of a unified 'dynasty' for any reasons other than their own propaganda!

I've picked two 'clean' examples of when a ruling family was disrupted and restored, and where it's hard to argue that they weren't a direct family descendant.

The Stuart rulers of England & Wales, Scotland, and Ireland* were interrupted during the Interregnum from 1649 to 1660 - when Oliver Cromwell ruled as Lord Protector. Charles I was executed by the Parliamentarians at the end of the Civil War/Wars of the Three Kingdoms, and his son went into exile before returning 11 years later as Charles II.

In France the Bourbon monarchs had a restoration following the defeat of Napoleon in 1814, and then again in 1815 following the defeat of Napoleon. Or it was one interrupted restoration, it depends on your interpretation. You could also claim that Napoleon's heirs returning to rule in France was a counter-restoration, but it wouldn't really add any value to the historiography.

Edit: I don't know if this will help your netball argument, but I hope it answers your question.

Sure. I mean the "unbroken lines" contain "oops no son or close relative, the Dowager/regency best find a suitable successor pronto". Or "ok so I may have just killed the Emperor. Find me a relative to bring in". Dynasties have survived no close heirs, abdication, regicide, what is a little dynastic fall to stop one? Ok, quite a lot but it certainly wasn't an automatic stopper.

u/OldBoatsBoysClub mentioned Chinese dynasties so I will mention one. The Han. 400 years rule from 202 BCE – 9 CE, 25–220 CE. You may notice a bit of a gap there. The first part starts with Liu Bang (also known as Emperor Gaozu), his line ruling from Chang'an and sometimes referred to as the Former/Western Han. The second part is from Liu Xiu (also known as Emperor Guangwu) who set up his capital at Luoyang for what is known as the Eastern/Later Han. The gap in between was the eunuch regent turned Emperor Wang Mang and his Xin dynasty, though his reign was short and he was seen as a usurper, some of his changes would be adopted by the Later Han.

Were they the same dynasty? They felt so, that this was a continuation of the same dynasty. Liu Xiu could claim descent via Emperor Jing, grandson of the founder of the Former Han who had died in 141 with many many children. The Later Han as an empire drew upon legitimacy from the Former Han and during its darker days near its end, people talked of a four-hundred-year dynasty rather than a two hundred-one. Things, like the change in capital, personal, claiming to be more restrained and the abolishment of the role of Chancellor, might be different but it was still the Liu clan ruling, that had changed as they dealt with the aftermath of the civil war and restoring authority. But it was still the Han, Emperor Gaozu was still sacrificed to and Chang'an was an important symbolic city, the power of Fire as the Han's element was an attempt to be a continuation of the Han legitimacy rather than a break from, and the Han continued.

Those that followed and historians even now accept the 400 years, that yes there was that brief interruption of the Xin and civil war but the Han, via a different line, had continued till Liu Xie, Emperor Xian, abdicated on 11th December 220. Now terms like Former/Later, Western/Eastern do act as a separator but that isn't an attempt to delegitimize the idea of the Han's continuation. Nor to go "actually that second Han only lasted 195 years so all that stuff about 400 years, continuation and so on is rubbish" as 400 years is still used. However, it is useful to be able to separate which part of the Han one is talking about since there were differences and which one someone might have expertise in (400 years is a long time).

I should also note that in the civil war that followed the collapse of Han authority, one of the factions in play called Shu-Han claimed to be a continuation of the dynasty. Since Emperor Xian abdicated (and, initially, they claimed was killed), the mandate passed to the last Liu standing in Liu Bei (Emperor Zhaolie). He claimed (distant) descent via Emperor Jing and had before self-declared as King of Hanzhong to draw legitimacy via Emperor Gaozu's old title, they didn't consider themselves new claimants to the mandate but restoring the rightful mandate and continuing the Han. That the Cao's were just another set of Wang Mang's, a disruption but not a end. As he and his son Liu Shan (Emperor Huai) didn't unite the land, nowadays it isn't considered the third era of the Han, the Han as the unified force dies with Emperor Xian's abdication to Cao Pi.

But in the debates about who had the mandate between Emperor Xian abdication and the unification of the land under the Jin dynasty of the Sima clan, Shu-Han's claim gave them legitimacy in the centuries after. Eastern Jin historian Xi Zuochi revived the argument that the mandate had gone across to the fellow Liu's before going to the Sima when they conquered Shu-Han in 264. Though that didn't take off at the time, it took hold during the Song dynasty with other scholars painting Shu-Han as the legitimate successor to the mandate, the rulers of Han Zhao would sacrifice to Liu Bei and Liu Shan as part of the Han dynasty. That descent and continuation of the Han is part of Liu Bei's cultural popularity with the novel Romance of the Three Kingdoms really making play with his link to the Han. Sure his claim might be based going back generations upon generations (via Emperor Jing's son Sheng who had 120 children), of the Former rather than Later Han but it was enough for Liu Bei and his followers to make that claim and for it to give a claim to the mandate in the centuries since.

The fall of dynasties and the civil wars that came with it was extremely disruptive, the claimants were distant relatives who were shaped by the chaos, and the dynasties they created were far from exact replicas as they themselves adapted. Yet for all the change, they saw themselves as a continuation, in the case of the unifying Han's that is still very much accepted. The Han ruled for 400 years a unified land, with a break in the middle and a new branch taking over, but still the Han.

So, I would copy and paste something that I wrote in response to a question in the Tuesday Trivia post about whether the CK2 matrilineal marriage that allows women to pass on dynasty names was a real thing:

Not really; the rigid mechanic of CK dynastic family tree is itself a total anachronism. The concept of a "dynasty" is itself more about historians sorting and organizing families than something people were deeply concerned with at the time - it's just a surname, really. The idea that women couldn't possibly inherit a throne or, more usually, pass a throne to their descendants was not common, so there was no need for special marriages designated as "matrilineal". I have a number of past answers that are relevant to this:

Why did Londoners reject Empress Matilda in 1141?

Cadet Branches (inspired by Crusader Kings 3)

When Isabella and Ferdinand joined the houses of Castile and Aragon, they ruled as practically equals. Was it unusual for a queen to wield such political power and influence in 15th Century Europe; and what did contemporaries write about the extent of Isabella’s power and influence over Spain?

On Berenguela of Castile

When it comes to European history, the concept of the dynasty has generally been of greater importance to historians and pop culture as a way of differentiating different periods than to the people actually making up said dynasties. Historical monarchs cared about perpetuating their families' reigns, they weren't obsessed with continuing their surnames or "house" names. When Juana I succeeded Isabel of Castile, bringing the Trastamara dynasty to an end and replacing it with the house of Hapsburg, her husband's family name, this was still considered the continuation of a bloodline. When Philippe of Valois succeeded Charles IV of France, his cousin, the name of the dynasty changed but nothing was perceived as being lost.

It's not that common for dynasties to "reform" simply because most of the time when they changed, there were no people left from the original dynasty to carry on the name (such as when Juana succeeded her mother - there were no sons left among Isabel's children) or because the family had been ousted from power and replaced (such as when Henry (VII) Tudor usurped Richard (III) Plantagenet, subsequently imprisoning and executing pretty much all remaining Plantagenets). So an excessive need for similarity when comparing to sports teams simply doesn't work, because every team is itself like its own dynasty, different rosters being like different monarchs, with a continuation of the name despite changes in lineup. Only the end of a team's existence would be comparable to the end of a dynasty, if you're being as literal as pahamack.