When looking at the path of economies over time, you see recessions and recoveries. But there also a well-known is a common cognitive pattern of "paraedolia," which refers to looking at randomness and perceiving patterns that aren't really there. Even using the conventional term "business cycle" for patterns of recession and recovery hints at a belief that the economy is be based on underlying patterns and dynamics that will cause it to rotate in a preordained way from recovery to recession and back again. When people ask whether the recovery is "getting old" or has has gone on "long enough," they are presuming this kind of "cycle."
The statistical answer to whether economic upswings die of old age can be answered statistically, and Glenn D. Rudebusch summarizes the conventional wisdom very nicely in "Will the Economic Recovery Die of Old Age?" written as the Federal Reserve Bank of San Francisco "Economic Letter" for February 8, 2016. Rudebucsh uses a kind of graph called "survival analysis," which can be applied to people's chance of dying, to part of a machine wearing out or breaking, to whether economies fall into recession, and many other applications.
As an example, here's a survival curve for the probability of an American male dying in the next year, The graph shows that the chance of dying in the next year doesn't rise very much at all for men up to the age of about 50 or 60, but then it starts to rise steadily with age.
A survival curve for the economy asks a question like: "What's the chance of an economic recovery ending in the next month?" Based on data for US business cycles going back to 1858, the patterns look quite different for before and after World War II. Here's Rudebusch's figure. Before World War II, there was a substantial rise in the chance of recession as an expansion aged: that is, after about four years, the chance of a recession int he next month has reached 20% and climbing. But since World War II, the chance of a recession rises by comparatively little as a recession ages: it's maybe 2% chance of recession in the next month after four years, but still only a 4% chance of recession in the next months after 10 years.
In short, US business cycles in the last 70 years or so don't seem to have a natural lifespan.
However, the notion of predictable cycles was once very hot stuff in the economics profession. One classic exposition is the great economist Joseph Schumpeter's 1939 book on Business Cycles (an abbreviated version is available on-line here). Schumpeter suggested that the rise and fall of the economy could be understood through a mixture of three different kinds of cycles: short-run, medium-run, and long-run. The short-term 3-5 year cycles were called Kitchin cycles, and Joseph Kitchin argued in 1923 that they were based on variations in of psychological factors and crop yields. (If you want more, see Joseph Kitchen, "Cycles and Trends in Economic Factors," Review of Economics and Statistics, January 1923, 5:1, pp. 10-16.) The medium-run Juglar cycles were based on fluctuations in levels of fixed investment often stemming from waves of innovation, as first argued in an 1869 book by by Clément Juglar (a condensed English translation is available here). The long-run Kondratieff cycles happened every 50 years or so, give or take a decade or two, and were based on major technological shifts (an English translation of Nikolai Kondratieff's 1922 article is available here). For example, Schumpeter suggests in his 1939 book that a Kondratieff cycle had run from the start of the Industrial Revolution in the 1780s up through 1842, when it was followed by what he called "the age of steam and steel" from 1842 and 1897, and then age of "electricity, chemistry, and motors" after about 1898.
But as Schumpeter was quick to note, the idea of three overlapping cycles wasn't meant to be definitive. He wrote: "There are no particular virtues in the choice made of just three classes of cycles. Five would perhaps be better, although, after some experimenting, the writer came to the conclusion that the improvement in the picture would not warrant the increase in cumbersomeness."
For the modern economist, this notion of maybe three or maybe five overlapping cycles, happening over maybe 3-5 or 7-11 or 40-60 years, sounds a lot like an attempt to impose an overall template pattern that isn't really there on an essentially random set of events. Sure, one can look back after the fact and analyze the proximate causes of recessions, like the Federal Reserve raising interest rates to fight inflation in the early 1980s, or the aftermath of the dot-com investment boom in the later 1990s, or the housing price bubble leading up to the Great Recession. But those proximate causes were not an inevitable cycle; instead, they were the result of other economic events and policy choices.
So the good news is that the US economy doesn't seem to be doomed by any mechanical law of aging recoveries to enter a recession soon. After all, there was a period between recessions in the 1960s that lasted 106 months and the another period between recessions from the 1990s into the early 2000s that lasted 120 months. But on the other side, the US economic recovery is far from bulletproof, and remains vulnerable to twists of policy and fate.