A variant of the CRT that has many practical applications is the "explicit CRT" (see http://cr.yp.to/antiforgery/meecrt-20060914-ams.pdf, for a good introduction).
One scenario where the explicit CRT (mod m) can be used, is the following. Suppose we know the value of an integer x modulo primes p_1, ..., p_n, whose product exceeds, say, 4|x|. Now we could use a standard CRT computation to determine x, but suppose we actually wish to know x mod m, where m is some integer not among the p_i. The explicit CRT lets us compute x mod m using arithmetic operations that involve operands that are all about the same size as m (rather than x). This is useful when x >> m; if x is a 100 digit number but the p_i's and m are all 3 or 4 digits numbers, you could compute x mod m on a pocket calculator with an 8 digit display, given the values x mod p_1, ..., x mod p_n.
This technique is especially useful when one needs to compute many large integers x modulo the same m (the integers x might be the coefficients of a polynomial and m might be the characteristic of a finite field, for example). This approach is used in http://arxiv.org/abs/0903.2785 to compute Hilbert class polynomials, and in http://arxiv.org/abs/1001.0402 to compute modular polynomials, both of which are notoriously large but can be efficiently computed with the explicit CRT.