No, what you say is not true. There are a number of factors that prevent required background from building up beyond reason. I can only speak for theoretical physics, but most reasons should hold for other sciences as well.
The first reason is scientific revolution. Periodically there will be dramatic paradigm changes that open up huge new areas of exploration. Even well-established facts become fresh again, as scientists need to explain them anew in the new theory. In the 1800's, celestial mechanics was an intensely mathematical field communicated through thousand-page monographs like Delauney's Théorie du mouvement de la lune. By 1920, it was unclear how to even describe planetary motion at all starting from the Schrodinger equation.
The second reason is technological progress. Staying with the example of celestial mechanics, many ingenious and complex mathematical techniques were invented to reduce problems "to quadratures", i.e. to a set of single integrals whose values could be found in tables. Nowadays almost nobody is taught these techniques, not even astrophysicists, because numerics has largely replaced them. The old methods still work, but they're not easier, so they're not emphasized.
The third reason is experimental invalidation. For instance, through the 1990's and 2000's, many increasingly complicated models of low-energy supersymmetry were devised, in the hopes that they would be confirmed at the LHC. They haven't been, and many of them have been outright ruled out. That's a few hundred papers I don't have to read.
The fourth reason is pedagogical improvement. We still teach students the material in Newton's Principia Mathematica, but the core material of this dense tome, once understood by only a few mathematicians in all of Europe, has been condensed into carefully edited textbooks, and disseminated in blog posts and Youtube videos. In general, 90% of everything, even masterpieces, is cruft that can be condensed away. For example, Newton insisted on using pure Euclidean geometry for many of his proofs, many of which are much shorter in modern calculus (which also now has much better notation!).
The fifth reason is increasing specialization. There once was a time when a physicist was expected to be able to understand just about any paper in physics, and that time was over 100 years ago. These days the Physical Review is not one journal, but over ten. The ArXiv has over 25 categories, with most people not even following all the literature in one. This might have been what you were getting at, but the point is that things have evolved gradually to keep requirements reasonable. Nobody requires a Ph.D student to understand every subfield, because that would indeed take forever.
The sixth reason is hindsight. This is related to the other ones, but distinct. Often research is only hard because there's a haze of confusion hanging around a new topic. The pioneers of quantum mechanics and quantum field theory wrote lots of papers that weren't wrong, but were quite confused. (And that's not to insult them; we are still confused now, just about different things.) They weren't sure about many things we take for granted today, and spent a lot of time checking things that are obvious in the modern formalism. So the energy needed to read one of their papers now is quite a bit less than was required right when it was published.
And the final reason is irrelevance. At the end of the day research is a human activity. If your theories become too complicated for anybody to understand, they just simply won't be read. Research is not a linear thing where everybody takes turns putting a brick on top of the tower; it's a branching tree. If nobody wants to build on your theory, if it's too complicated for the benefits it provides, they'll just do something else. (I will not name examples.) Suffice it to say there are many things working against the 'inevitable' trend you see here.
