How often do techniques/methods that are considered old or obsolete become relevant again? In many, if not most, disciplines methods age, get old and eventually become more or less obsolete, in the sense that some other method perhaps using a different technology becomes common-practice for a particular type of experimentation. One such example could be Edman sequencing, which used to be the method for sequencing peptides, but is pretty much outdated nowadays, not because it does not work, but because it's costly in time and labor.

In my experience, these old/obsolete techniques are taught both at Masters and PhD level, primarily to give a deep understanding of the field, albeit being more theory than practice. Are there any other benefits in learning/studying older methods, rather than only studying the state-of-the-art?

I was wondering how often, if at all, such old methodologies come back (perhaps following a breakthrough in overcoming some original limitation) and regain popularity. Are there known examples? 

As a follow-up; is such change in methodologies a phenomenon observed in experimental research, or are there equivalent scenarios in theoretical research as well?
There are some important examples of techniques and problems being "revived" after having lain dormant for many years. For instance, in my own field, one of the cornerstones of computer modeling was originally derived in the 1920's, and then basically remained undeveloped until the 1970's, when it was brought back in fashion. (It has since been replaced by faster methods, but it is still regarded as the "touchstone" for high-accuracy calculations.)

Similarly, entire fields of research may have been developed before the applications caught up with them. Again, bringing up an example from my own field of research, the class of materials known as ionic liquids was known as far back as the early 1900's, but again were forgotten until the 1990's (or so) when they were "rediscovered" because they were nonvolatile materials and could be used as replacement solvents in the chemical industry. (It also offered the ability to do "combinatorial chemistry," which was also coming into fashion.)

THe point of these examples is that sometimes theoretical and even experimental techniques may be developed in one era, but the ability to exploit the technology to its fullest extent has not yet been attained. Keeping awareness of such methods thus can have some use. Whether it should be taught to every student of a field is a different issue.