If an encryption system allows for character-per-character cracking, then it is awfully weak, and should not be used.
Mathematically, block ciphers are defined as pseudorandom permutations. A block cipher works over the space of blocks of length n bits; such a space has size 2n. There are 2n! permutations over that space (that's a factorial, meaning that the number of possible permutations is huge). A secure block cipher is such that it is indistinguishable from a permutation selected at random, uniformly, in the space of possible permutations: each key is supposed to correspond to such a random choice of permutation, and, crucially, all choices for all possible key values are independent of each other.
What this means, in mundane words, is that for a secure block cipher, either you have the whole key, exact down to the last bit, or you have nothing. Contrary to Hollywood depictions, an "almost good" key does not result in a "blurry plaintext": if even one key bit is wrong, you should get random junk (that is, an output which is sufficiently indistinguishable from random junk that you cannot know whether you are close to the right key or not).
Of course, if the encryption system is weak, anything goes. @Mark cites a case of "splitting" (in WPS) which is an atrocious weakness that can, indeed, be exploited for faster attacks. Padding oracle attacks also work over a byte-by-byte leak which allows for byte-by-byte reconstruction of the plaintext (not the key, but still).
Another, more technical example, is the old PKZip stream cipher: an "homemade" stream cipher which turned out, with all the unavoidability of Death in a Greek tragedy, to be weak; the stream cipher relies on several internal "keys" which can be unravelled one by one. I encourage people interested in cryptography to study that example, because it demonstrates quite well the way a cryptanalyst thinks, and why mere accumulation of operations does not guarantee security; and the attack is light enough to be implemented in practice (total cost is around 238, which is within range of a few hours of computation on a PC with decent programming, not necessarily optimized assembly).
A lot of cryptosystems from before the computer era were breakable on a per-character basis, because they had to be executed by the human brain of the operators, and such tools are not good at using large values or doing a lot of operations. The classical transposition and substitution ciphers (a very large family) tend to fall to character frequency analysis, which is, indeed, a per-character break.