I hear this line pretty frequently. But the data do not support this statement. One of the possible explanations of the belief that older products were more reliable is selection bias.
Products have a variable lifespan. Hence, when we use a century-old item, the item does not break easily, because it was stress-tested for a century and the weaklings were weaned a long time ago. On the other end, when we use a new item, there is a good chance that it won't survive long because it wasn't stress-tested and weaned for a century like the old products that survived to these days.
Another factor is the variable variance of durability. When the distinct count of the manufacturers that produce the product is high and when the manufacturers are independent of each (e.g.: they use only local resources), we may expect high variance in the durability of the products. On the other end, if there is just a few manufacturers or if they all use the same components or design, we may expect low variance in the durability of the products. Hence, some of the products that were produced during the high variance period are likely going to have outstanding durability (just like it is likely that some of them had really terribly low durability). The "issue" with the new products is, that we currently live in a fairly globalized world and many technologies that we use daily were commoditized (standardized and made widely available). Hence, many new items that we use daily have a fairly predictable lifespan. A lifespan, which does not span centuries (because that would be overkill). Consequently, we may sometimes find "indestructible" items that were created when the technology was new. But keep in mind that just like "indestructible" items were produced, there were also "rubbish" items, which were quickly thrown out.
Overall, the data suggest that the quality of manufacturing keeps improving over time. But thanks to selection bias and variable variance, the reverse appears to hold for the item-user.
Addendum: We could model it analytically. For simplicity, assume that probability of a product failure follows log-normal distribution (I picked this distribution because it fulfills three basic properties: it does not allow negative values, it has a long tail and people are familiar with it).
Selection bias can be then illustrated as a difference between probability that a new product fails in the next 10 years vs. a probability that a 100 years old product fails in the next 10 years. The first probability is going to be large, because log-normal distribution is "fat" at the beginning. But once we get to the tail, the derivative of the distribution is going to be close to 0. In other words, if a product survives 100 years, it is actually more likely that it will fail after 10 years than during the next 10 year period.
The variable variance can be illustrated with an observation that whenever an engineer doesn't know how to accurately estimate something, he/she prefers to overestimate the parameters and build the thing robustly. However, sometimes the initial design has a flaw, which reduces the lifespan of the product (hence, the peak is wide - the product can last long but also fail quickly). The next phase is fixing these flaws. But everything else is left as before (the fat from the beginning is removed but the fat tail is preserved - this is the period from which we observe many "eternal" products). Over the time, the product is price optimized (the fat tail is removed - the products have a predictable lifespan without extremes and they all look like garbage in comparison to the eternal products of the past).
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