“It is on the doctrine of the illusive nature of matter, and the infinite divisibility of the atom, that the whole science of Occultism is built.” – H.P. Blavatsky, The Secret Doctrine, I, p. 520
When HPB wrote this in 1888, the atom was conceived as an indivisible billiard ball of sorts, and now dozens of sub-atomic particles have been discovered, seemingly to the limits of mathematics and practicality. The recent Higgs-boson particle required a multi-billion dollar machine and wide international cooperation to discover.
“We are at the very close of the cycle of 5,000 years, of the present Aryan Kali-yuga; and between this time and 1897 there will be a large rent made in the Veil of Nature, and materialistic science will receive a death-blow.” (Blavatsky, 1888, SD I, pp. 611-12 )
Science was very complacent in 1888, having established Newtonian physics firmly, and it seemed only some dusting up around the edges was needed to have a full understanding of Nature. As HPB predicted, between 1888 and 1897, there was the discovery of X-rays by Rontgen in 1895, the natural radiation of Uranium in 1896 by Becquerel, practical radio by Marconi and others in 1897, and the establishment of the first sub-atomic partical – the electron, by J. J. Thomson in 1897. This all cascaded in many more discoveries in a few decades, and eventual proof that the sub-atomic world did not operate according to Newtonian physics.
In Astronomy, one of Blavatsky’s Eastern Teachers in a letter to A. P. Sinnett (circa 1885) wrote:
“Science will hear sounds from certain planets before she sees them. This is a prophecy .” 
This was before the birth of radio-astronomy, and a mysterious statement at a time before even the birth of radio in 1897.
On the nature of matter one of Blavatsky’s Teachers wrote:
“Western Science has still three additional states of matter to discover.” (Mahatma Letters, p. 341)
The three states of matter known in the 1880’s were solid, liquid, and gas. A fourth state of matter is now known and labeled “Plasma.” Plasma is created when a gas is heated above 2000 C resulting in electrons escaping the molecule and creating a highly ionized or polarized gas. Plasma thus responds to a magnetic field and is likely the “magnetic matter” referred to by K.H. as existing in the Sun (MLs pp. 163-5). Plasma is formed in fusion reactions, which is why controlled fusion reactions are difficult to maintain, as the plasma will melt any container it comes in contact with and must be suspended by magnetic field.
In Geology in HPB’s time it was known that there was a ridge of undersea mountains running the length of the Atlantic, but it was not known that this ridge extended up into the Indian Ocean until the surveys of the International Geophysical Year in the 1950’s.  In 1888 she wrote:
“. . . . the fact that the ELEVATED RIDGE in the Atlantic basin, 9000 feet in height, which runs for some two on three thousand miles southwards from a point near the British Islands, first slopes toward South America, then shifts almost at right angles to proceed in a SOUTH-EASTERLY line toward the African coast, whence it runs on southward to Tristan d’Acunha. This ridge is a remnant of an Atlantean continent, and, could it be traced further, would establish the reality of a submarine horse-shoe junction with a former continent in the Indian Ocean.” (Secret Doctrine, vol. II, p. 333)
Blavatsky made several predictions in the 1880’s that there would be widespread war in Europe in the 1900’s:
“It is simply knowledge and mathematically correct computations which enable the Wise Men of the East to foretell, for instance, that England is on the eve of such or another catastrophe; France, nearing such a point of her cycle, and Europe in general threatened with, or rather, on the eve of, a cataclysm, which her own cycle of racial Karma has led her to.” (Blavatsky, Secret Doctrine I, p. 646)
“Saint-Germain was certainly the greatest Oriental Adept Europe has seen during the last centuries. But Europe knew him not. Perchance some may recognize him at the next Terreur, which will affect all Europe when it comes, and not one country alone.” (Blavatsky, Theosophical Glossary, “St. Germain,” p. 309) 
On the origin of the predictions H.P.B. wrote that they weren’t “psychic” but based on a science of cycles:
” . . . . It is the knowledge of the natural laws that make of seven the root nature-number, so to say, in the manifested world – at any rate in our present terrestrial life-cycle – and the wonderful comprehension of its workings, that unveiled to the ancients so many of the mysteries of nature. It is these laws, again, and their processes on the sidereal, terrestrial, and moral planes, which enabled the old astronomers to calculate correctly the duration of the cycles and their respective effects on the march of events; to record beforehand (prophecy, it is called) the influence which they will have on the course and development of the human races … (S.D. I, p. 621)
Theosophist and Vedic scholar Subba Row made some predictions that perhaps have occured and some not occured yet:
“We are at the end of a cycle – geological and other – and at the beginning of another. Cataclysm is to follow cataclysm. The pent up forces are bursting out in many quarters; and not only will men be swallowed up or slain by thousands, ‘New’ land appear and ‘old’ subside, volcanic eruptions and tidal waves appall; but secrets of an unsuspected Past, will be uncovered to the dismay of Western theorists, and the humiliation of an imperious science. This drifting ship, if watched, may be seen to ground upon the upheaved vestiges of ancient civilizations, and fall to pieces. We are not emulous of the prophets honours; but still, let this stand as a prophecy.”
H.P.B. wrote somewhere to the effect that “the 20th Century may be the last by that name” and we have perhaps narrowly escaped the possibility of the destruction of our civilization during the Cold War with its poised Nukes around the world. On our future in this perilous age she wrote:
“If Theosophy prevailing in the struggle, its all-embracing philosophy strikes deep root into the minds and hearts of men, if its doctrines of Reincarnation and Karma, and Responsibility, find a home in the lives of the new generations, then, indeed, will dawn the day of joy and gladness for all who now suffer and are outcast. For real Theosophy IS ALTRUISM, and we cannot repeat it too often. It is brotherly love, mutual help, unswerving devotion to Truth. If once men do but realize that in these alone can true happiness be found, and never in wealth, possessions, or any selfish gratification, then the dark clouds will roll away, and a new humanity will be born upon earth. Then the GOLDEN AGE will be there, indeed.
“But if not, then the storm will burst, and our boasted western civilization and enlightenment will sink in such a sea of horror that its parallel History has never yet recorded.” 
– M. R. J.
 The Mahatma Letters to A.P. Sinnett, Theosophical University Press, p. 170
 Theosophical Notes, Victor Endersby, No. 1, 1961, p. 6
 The Theosophical Glossary (Theosophy Company, Los Angeles) was published posthumously and not all written by Blavatsky, so there’s some doubt on the authorship of this quote.
 T. Subba Row Garu, (1856-1890.) This quote was from a Theosophical publication, but I haven’t tracked down the original source for it.
 “Our Cycle and the Next,” first printed by H.P. Blavatsky in Lucifer for May, 1889.
General Reference: “H.P. Blavatsky and the New Physics,” Boris de Zirkoff, Theosophia, 1947; “Theosophy and Science,” M.R.J., Protogonos, Fall, 1989; “Our Thoughts, Our Earth,” Richard Robb, Protogonos, no. 19, Jan., 1995