Are there any known attacks that try to use statistical models in this way to break encryption?
Yes. As gowenfawr noted they are typically effective against an older class of encryption algorithms called monoalphabetic substitution ciphers. In these ciphers a letter in the alphabet is exchanged with a symbol. The same letter is always substituted with the same symbol. For example replace every a with i and replace every b with l and so on. This simple substitution preserves the statistical nature of each letter. To attack a monoalphabetic substitution cipher you just need to know the original language used and the statistical frequency of each letter. Find the most common symbol in the cipher text and replace it with the most common letter in the original language. For English the most common letter is e.
The next progression from monoalphabetic substitution is polyaphabetic substitution. This type of cipher uses a table of alphabets to choose a symol to substitute for a plain text letter. This may effectivly blunt a statistical attack (if the key is long enough) by making the substitution a function of the key and the letter instead of just the original letter.
More modern algorithms make the encipherment of each letter a function of many inputs which makes detecting a statistical pattern much harder. One of the input is a key which is usually a piece of random data. Using randomness as an input blunts the ability of a cryptoanalyst to detect a pattern.
However statistical analysis can still play a role in analyzing encrypted data. This usually comes in the form of what is called a side-channel attack. A side channel is a transmission of data that is a side effect of the transmission of the encrypted data. A recent paper "Uncovering Spoken Phrases in Encrypted VoIP Conversations" describe how variable bitrate encoding produced a side channel for the encrypted voice data.
Variable Bit Rate Encoding
When sound is converted into data it becomes bits of information. Not all words and sounds require the same amount of data to represent them. One way of transmitting the voice data is constant bit rate. FThe voice encoder (not encrypter) translates the sounds into data which is then encrypted and transmitted. For constant bit rate every second of sound produces the same amount of data. If the sounds made during that second produces less bits than need to be transmitted then the encoder fills the remaining bits with silence. So, constant bit rate has a lot of meaningless data. Variable bit rate allows the encoder to send as many bits as were generated by the sound by the size of each sample/frame of sounds must be indicated by the data. Some encryption algorithms produce encrypted data that is exactly the same size as the plaintext data. If you use a length preserving encryption algorithm with variable bit rate encoding then an attacker can see the length of each sound sample and sucessive sample lengths and use statistics about the way people talk to pick out a few phrases.
Are they effective? What are their short-comings?
Statistics can be effective in special cases. The sortcoming of statistics is that they tend to be limited to speach and plain text which are a smaller component of communications today then they used to be.