If I had to hazard a guess, this is an artifact of conversion from fractional dimensions, with all the 'significant' digits left in.

A competent engineer working with this will just chuckle and truncate to appropriate working 'precision'...

Bingo. Working with injection mold ejector pins, which are still manufactured to nominal fractional sizes, I recognize all those four place decimals as conversions of some multiple of 1/32 on an inch.

I think that the New Haven only specified to 32nds of an inch, .03125". These listed dimensions are oversize by .00005", or one half of a ten thousandth of an inch:
Web - 0.5938 should be .59375" or 19/32"
Fishing - 3.3438 should be 3.34375" or 3-11/32"
Depth of Head - 1.7188 should be 1.71875" or 1-23/32"

By the way, the Elevation of the bolt hole for this section is 2-39/64" or 2.609375", and for 100 pound American Railway Association Type B rail, a standard design: 2-65/128"or 2.5078125", that is, to the ten millionth of an inch for those who wish to torture themselves with decimals! My source is the table on p. 35 of the 6/79 L. B. Foster Co. catalog that I cited earlier.

" to the ten millionth of an inch " Gee, guess I'll have to buy my track crews a micrometer so they can drill 'em in the right location. To think we've been using a tape measure all these years. ;)

A few years ago I royally annoyed a designer who had drawn prints for the track installation to 3 decimal places of a foot (instead of the usual two). At the pre-construction meeting I asked "Are you serious about this?" He earnestly replied that he was... I then said "OK, so what temperature will you be measuring them at?" "Uh, huh, what?" "Well, the rail expands and contracts far more than that, so I guess I need to know what temperature this was designed for..." He later begrudgingly admitted that maybe working to the 100th of a foot would be sufficient. (In most cases, the closest 10th would be fine.)

What had really happened? Well, they all design them to many decimal places, he just hadn't bothered to tell his CAD program to round them and show the resulting two decimals. But he didn't want to admit he'd made an error.

So tell me, who in the days of slide rules calculated out that spec? Three significant figures, indeed!

Well, the who would be the American Railway Engineering Association, or AREA. For those not familiar with AREA, since merged into AREMA, it's a group of engineers (civil, electrical & mechanical type) from railroads and industry that work together to design things like rail, ties, OTM and the like as well as anything else related to railroad systems. The full AREA manual is huge, four volumes measuring nearly 18 inches wide. The four volumes cover track, structures, infrastructures and management. If you were taken to some foreign country and told "You must design a railroad system for our country. You have to do it on your own. You can buy one book...", this would be the book to buy.

What's more interesting, at least to me, is the "Why?"

65/128. That's not really to the millionth, it only works out that way when converted to decimal form. In fractional form, it's 1 part in 128. Still, it's an awfully precise number, as noted, three significant figures. It's also very close to a rather imprecise number, 64/128, better known as 1/2 inch.

Does the additional 1/128th of an inch really matter? Especially when we're talking about a bolt hole that has some slop built into it by design. Did they really want it to be that precise? Did they possibly not like the implied inaccuracy of saying 1/2 inch, and so said "OK, so what's a hair more? 65/128, that will tell them we want it to be precise!"

I have to expect you couldn't get that kind of accuracy when drilling holes in the field in those days. For that matter, we don't work to that kind of accuracy today. Many of our bolt holes are drilled using a template, but I certainly wouldn't bet on it being so precise that it's within 1/128th of an inch.

Just because a dimension is on a drawing doesn't mean that the article was manufactured to that dimension. In fact, dimensions in 64ths are fairly rare in RR drawings. Engineers with any real world experience will always specify an allowable tolerance from the "standard" to allow for manufacturing or field work. Thus, that 65/128 dimension may actually be acceptable from 61/128 to 69/128 given a tolerance of 1/32 of an inch. The half inch hole may be as small as 15/32 or as large as 17/32 and still be perfectly acceptable though I suspect the tolerance here would all be on the plus side since the bolts are seldom undersize. Engineers would specify much smaller tolerances for items coming out of the machine shop where precision could be closely controlled.
The fact of the matter is that since electronic calculators have become the standard we, especially the non-engineers or scientific types, have lost track of the difference between accuracy and precision. The guys with the slide rules were always faced with problems in precision since they were forced by their tools to be always aware of significant digits.
And God help us when the LAWYERS get involved since they are trained to be hair splitters and know nothing at all about tolerance!

I'll ask the opposite question... Why deliberately make it imprecise? If I recall, this is the dimension from the base to the bolt hole centers. The design criteria would have been to center these in the fishing dimension, which in turn was driven by other factors in the design of both head and base. Someone just worked the math, and rounded to the smallest division commonly used on machinist scales of the day, 1/128", knowing full well that the precision was smaller than the tolerance, but preserving the design intent that the holes be centered on the joint bars.

I don't see a problem. The problem would have been if the general tolerance on the hole location was +/- 1/128".