Swadesh's idea of a constant rate of word evolution of the basic vocabulary items led to the development of the field glottochronolgy, a field of linguistics which attempts to date the most recent common ancestor between two languages by comparing cognates from Swadesh lists. The first attempt to accurately date a language split was by Swadesh himself in the paper Salish Internal Relationships,  to attempt to discover the phylogeny of the Salish languages. (The Salish languages, now critically endangered, is one of the world's oldest language and  is spoken by Native Americans who pre-contact homeland was the area that is now southern British Colombia and Washington State). The existence of the family and their overall grouping had been known for some time, however the internal sub-grouping was still largely a mystery. Swadesh attempted to elucidate these subgroups by assembling 165 word long Swadesh lists of 30 Swalish languages, and performing pairwise comparisons on the percentage of cognates in each language, these was quantified by the following equation:
\(i\ =\ \frac{\log\ \left(c\right)}{2\cdot\log\left(r\right)}\)
Where is the percentage of shared cognates,  is the indicated period of depth in time (an arbitrary unit) and is the percentage of basic vocabulary retained after 1 period. was estimated in \cite{Swadesh1950} as 0.85 by comparing contemporary English to old English ( atime difference of roughly 100 years); this constant was formalised and re-estimated in 1953 by Robert Lees in The Basis of Glottochonrology \cite{Lees_1953}. Now coined the glottochronological constant, Lees re-estimated the constant by using a 1000 year comparison between 13 languages and generating a mean of the result: 0.80484±.0176 (with a 90% certainty). was then replaced by t, now indicating a 1000 year time difference, not an arbitrary unit. 
\(t\ =\ \frac{\log\left(c\right)}{2\log\left(r\right)}\)
The field of glottochronology was contentious since its birth, however did find popularity with researchers at the time. However Swadesh's underlying assumptions: The rate of loss of basic vocabulary items was assumed to be the same for all languages and the rate of retention of  basic vocabulary items was assumed to be constant over time are, at best, large oversimplifications. Cognates for pairs of languages can also be difficult to identify: Large sound shifts (example) and analogous morphemes (example) can lead to an inaccurate estimation of relatedness, and hence inaccurate split date. 
However the comparative method is not without flaws, or criticism: Ironically these issues fall into the camp of the comparative method being too similar to genetics, and issues caused by the comparative method not being similar enough.
In the former camp, many modern day linguists consider the tree model implied by the comparative method as overly simplistic, and express doubts that genealogical comparison like is misleading. A tree model implies a series of distinct nodes: It implies sudden and irrevocable change in a population preventing further contact ie many different proto-languages coexisting without further interaction, which is clearly misleading in areas of continuous landmass. Critics claim the model ignores situations where many dialects within a language evolve into distinct languages, over a long period of time where innovations are shared (known as areal diffusion). Populations bordering others have a higher borrowing of linguistic features of each other, as well as similar cultures - influencing linguistic change in a shared direction, this is known as linkage. Different models such as the wave model, which attempts to describe how language features spread over a continuous territory, and are gaining popularity with modern linguists as a way of modelling linguistic linkage and areal diffusion. A good analogy is of a pebble thrown into a pond, the effect is strongest near the epicentre, and weakens as it radially spreads. 
Another problem picturing languages as distinct nodes is the idea of linguistic uniformity within a proto-language, despite the existence of dialects within even small language communities (However the real life implications of this simplification are doubted) \cite{campbell2004}.  
The other main issues with the comparative method come from an underlying assumption that sounds will evolve in a fixed way; despite the reality that linguistic change is far more random and shares a lot of issues with genetics. A good example being analogy, where words of separate and unrelated languages coincidentally converge implying false relatedness. Words and linguistic features are often borrowed from other languages, which can imply a closer degree of relatedness that is correct.