Is it true that only the positions of the first and last letter in a word matter, the rest can be scrambled?

I read this sometime ago,

Aoccdrnig to a rscheearch at Cmabridge uinervtisy, it deosn't mttaer in waht oredr the ltteers in a wrod are, the olny iprmoetnt tihng is taht frist and lsat ltteer is at the rghit pclae. The rset can be a toatl mses and you can sitll raed it wouthit porbelm. Tihs is bcuseae we do not raed ervey lteter by it slef but the wrod as a wlohe.

I'm guessing that the Cambridge research probably doesn't exist, but is it true that we can normally understand a sentence whose letters are scrambled in this way?

Sort of, but not exactly. In addition to the languagehat post, here are the actual facts "according to a researcher at a Cambridge University". (That page is itself a summary of actual research on related matters; my summary will be inadequate.)

While it is true that (most) people don't read words letter-by-letter, and we can cope with some amount of jumbling, this happens through word shape and disrupting the order of letters can significantly affect the word shape. That is, it is not true that "The rset can be a toatl mses and you can sitll raed it wouthit porbelm". Rather, there are several properties of your paragraph that make it readable:

• In most of the words, the letters are only slightly transposed, not jumbled beyond recognition. For instance, "doesn't" is made "deosn't" rather than "dn'oset", "problem" is made "porbelm" rather than "pebrolm". If you try generating actually random transformations of the paragraph, you'll find it a bit harder to read.
• 33 of the 67 words are 1-, 2- or 3-letter words that are in exactly the right order, and another 12 are 4-letter words in which only a transposition is possible. The short words tend to be the function words so essential to meaning; paragraphs consisting of predominantly longer words would appear harder to understand.
• It is written in lowercase; if it were written in uppercase a lot more of the word shape would be lost. (This is also why uppercase is harder to read in general: all word shapes are rectangles.)

The page considers the sentence "The sprehas had ponits and patles". How would you read this?

The sherpas had pitons and plates.
The shapers had points and pleats.
The seraphs had pintos and petals.
The sphaers had pinots and palets.
The sphears had potins and peltas.

And many more.

It gtes mcuh hedrar wehn you hvae legnor wdros and you slpmiy rsrevee all the ioiretnr lrettes.

It is generally true, but I can think of some caveats:

1) It won't work well with all uppercase letters, as it is the shapes of the words that are our cues. All-uppercase words are essentially rectangles.

AOCCDRNIG TO A RSCHEEARCH AT CMABRIDGE UINERVTISY, IT DEOSN'T MTTAER IN WAHT OREDR THE LTTEERS IN A WROD ARE, THE OLNY IPRMOETNT TIHNG IS TAHT FRIST AND LSAT LTTEER IS AT THE RGHIT PCLAE. THE RSET CAN BE A TOATL MSES AND YOU CAN SITLL RAED IT WOUTHIT PORBELM. TIHS IS BCUSEAE WE DO NOT RAED ERVEY LTETER BY IT SLEF BUT THE WROD AS A WLOHE.

Still doable, especially having already read it once, but not as easy.

2) Following from #1, distinctive letters that give the words their shape should be in or very near to their "normal" location. If you notice from the text you posted, most instances of i, t, d, l, p, g, h, etc. appear at or within one character position from normal. The last word, "wlohe" does not, but instead swaps the tall l and h to give the word the same overall shape.

3) This works when the words are in a meaningful sequence, because our minds have a good idea of the subset of words that can follow any other, given the context and our understanding of syntax. We can further narrow it down by shape, and finally by letter, especially first and last. Out of context, these become anagrams, and suddenly rather more difficult to decipher.

No, it's true. Most people can process the words regardless of interior order. I read that paragraph almost as fast as I would normally, as I'm sure most people here did.