In biology, the lines between different groups in classification systems are rarely as sharp as we imagine them to be. This is because life is constantly evolving and changing. In a simple thought exercise, you can imagine that in the very distant past, the precursors of modern plants were all monocots. Dicot species did not emerge instantly from these precursors, as a distinct species from their monocot parents. That would be analogous to a pet lizard that laid eggs which baby birds hatched from. Rather, gradual evolutionary change causes numerous small differences that over time led to different morphology, and eventually making them distinct enough that we would classify them as different classes or species upon examination. However, many "in-between" stages may still exist as modern species as well. In the case of monocots and dicots, I found this website:

monocotdicot

The most direct answer to your question concerns these "in-between" species: "It is now believed that some of the dicots are more closely related to monocots than to the other dicots, and that the angiosperms do not all fit neatly into two cades."

That comes from a section on the fuzzy distinction between the classes: "Even after the general acceptance of Monocots and Dicots as the primary groups of flowering plants, botanists did not always agree upon the placement of families into one or the other class. Even in this century some plants called paleoherbs have left problems for taxonomy of angiosperms. These plants have a mix of characters which do not occur together in most other flowering plants. For instance, the Nymphaeales, or water lilies, have reticulate venation in their leaves, and what may be a single cotyledon in the embryo. It is not clear whether it is a single lobed cotyledon, or two which have been fused. The water lilies also have a vascular arrangement in their stem similar to that of monocots.There are also monocots which posses characters more typical of dicots. The Dioscoreales and Smilacaceae have broad reticulate-veined leaves; the Alismataceae have acropetal leaf development; and Potamogeton is one of several monocots to have floral parts in multiples of four. This "fuzziness" in the definitions of Monocotyledonae and Dicotyledonae is not simply the result of poor botany. Rather, it is a real phenomenon resulting from the shared ancestry of the two groups. It is now believed that some of the dicots are more closely related to monocots than to the other dicots, and that the angiosperms do not all fit neatly into two Cades. In other words, the dicots include a basal paraphyletic group from which the monocots evolved.

Wow, this is a difficult question.The terms monocot and dicot are words we have invented to separate plants depending on what kind of seeds they have (specifically, how many cotyledons are produced during one stage of development). Further, adult monocots typically have certain characteristics (such as vascular bundles arranged randomly) that differ from those of dicots (which, for example, have vascular bundles arranged in a circle). It is important, however, to keep in mind that the natural world does not always conform nicely to our expectations. This means that some plants that are dicots actually may be more closely related to monocots.

This means that there probably have been (or even are today!) plants that are not easy to identify as either a monocot or a dicot. They may have had characteristics of both, for example. In terms of the defining characteristic of how many cotyledons they have, though, you would have to think of a plant that had something between one and two cotyledons-- for example, by having one cotyledon and then part of another one. This is hard for me to imagine, but you never know, since it seems like with nature anything is possible!

In terms of cross-breeding a monocot with a dicot, or grafting part of a monocot together with part of a dicot, I do not think these things are possible since their development and vascular systems are so different. Hopefully a botanist or horticulturist can provide you with a more definite answer, though. And if they do tell you its not possible, keep in mind that it may not be possible right now, but some day somebody like you may find a way to do it!