The idea of isochrons is this: Suppose X is a parent element that decays in to Y and Z is another isotope of Y not produced by radioactive decay. Let x, y, and z refer to their concentrations. Since Y and Z are isotopes, we would assume they have similar chemical properties. Let's assume that initially, the ratio of y and z is constant, and then X begins decaying to Y. We have two sources of Y, so y = c1 * x + c2 * z at the end of some time period. It follows that the ratios x/z and y/z have a linear relationship whose slope yields the age of the sample. If these ratios are observed to obey such a linear relationship in a series of rocks, then an age can be computed from them.
However, we can imagine situations in which such a linear relationship could be produced without indicating a true age. Let A and B be two rocks containing only X and Y and no Z. Suppose A is very old (or appears very old) and B is very young. Suppose A and B become thoroughly mixed. Their perceived radiometric age would then be between that of A and B. Now, suppose a mixture of Y and Z penetrates this mixture of A and B, in some places more than in others, but with a constant ratio of Y and Z. This will then yield a beautiful isochron, but the age given will be meaningless. This can also happen if water removes a constant fraction of X but no Y from A, making A appear older, and then the mixture of Y and Z enters. Another possibility is for A to have a constant concentration of X and Y at the beginning, and for more Y to enter, making A appear older. Then if a mixture of Y and Z enters, a nice isochron yielding a false age will be produced. A final possibility is for A to have a constant ratio of X and Y at the beginning. Then a lot more Y enters by diffusion. Then the rock is heated and mixed so the ratio of X and Y is everywhere the same. This makes the rock look much older. Finally, a mixture of Y and Z enters, different amounts at different places. This will also produce a false, and much too old, isochron. I would say that these scenarios are not at all implausible, especially when one considers that the daughter element Y is often argon, a gas that is relatively mobile in rock.
For a much more detailed discussion of isochrons and a variety of possible problems with them, and how geologists attempt to solve them, see http://www.talkorigins.org/faqs/isochron-dating.html.