The most popular approaches to quantum mechanics would argue that the entanglements never end, they just get arbitrarily weak. In theories built on decoherence, the entanglements are always present, but they become less and less statistically important as they interact with a "heat bath" of particles in a random state. When they do so, the effect of these random interactions start to overpower the effect of the entanglement and the entanglement becomes increasingly negligible for predicting future measurements. When the entanglement accounts for 0.00000000000001% of the expected result of a measurement, the effect of this entanglement falls into the noise in our measurement errors and we can hand-wave it away.
What complicates this somewhat is that the major interpretations of quantum mechanics claim that a "measurement" causes this decoherence. This is because the purpose of the interpretations of quantum mechanics is to tie the world of quantum mechanics to a hypothetical world governed by classical mechanics. This connection is done through vague terms like "measurement" and "collapse."
What we find is that, in practice, the things called "measurements" can be implemented by thinking about them with the decoherence model. In a sense, we can use this way of thinking to realize the very abstract concepts of "measurement." They are typically built around using a great body of atoms in a known state (put there with energy in the classical sense), which then interact with a quantum system in a way such that the statistical expectation of the results looks like a measurement in the classical sense (like how one might measure a stick to be 12 cm long). These "measurements" do indeed have the effect of breaking entanglements in the way you describe, and the exact process of doing so is well described by multiple interactions with particles that are in a random state.
Something I found useful for working through this is the idea of weak measurements. While the exact meaning of a weak measurement is not universally agreed upon, it typically takes the form of doing an interaction with the system that falls short of being a "measurement," in effect creating an entanglement, and then taking the actual measurement later. Seeing how a weak measurement of an entangled system causes entanglements of its own is very helpful for seeing how these entanglements and measurements pile up.
