Of course you can picture 4-D space. Just picture n-dimensional space and then let n go to four.
Seriously, though, relativity tells us there's at least 4 dimensions, and quantum field theory tells us there's either 12 or 28. It's extremely common to consider physical problems in infinite dimensional spaces.
Recall that, by defintion, a dimension is just the least number of linerally independent coordinates required to specify uniquely the points in a space. For everyday life we usually only need to describe a position by it's spatial coordinate, which is why we're used to 3 dimensions. If you want to compare times, you treat time a fourth. You continue the process until you have the number of dimensions required to assign every possible "point" in the universe to an independent coordinate. When you determine this number, you know the dimension of the universe.
<math> OK, what I said above isn't really true, since spacetime is curved. Hence, physicists say that spacetime is a "manifold." This means that while we can describe everything in terms of orthogonal dimensions, what we really do is describe things in terms of locally euclidean manifolds which are homeomorphic to R^n where n is the lowest number of orthogonal dimensions required to describe all elements of the universe discretely.</math>
It's also possible to create more "artificial" spaces. For example, you can consider velocity as a dimension. Thus, a particle moving in 3d space will exist as a point in our new 6d space. (3 position coords, 3 velocity coords). As it turns out, this can make analyzing certain physics problems (like pendulums for example) much easier.
<math> Likewise, consider an n-degree polynomial, where n is a natural number or 0. It will the form a+bx+cx^2+dx^4+...+px^n. We can represent this as a curve in 2d space [ie x vs. f(x)] or we can represent it as a point in n-dimensional space. [with x^k being a unit vector in the k-dimension]. Proving orthogonality is left as an exercise to the reader.
</math>
Pretty much what people have been saying in this thread is true: Don't assume that because you can only immediately observe 3 (or 4) dimensions that that's all there is in existence. Dimensions aren't really "things"; they're really just measures.
If you find two particles that are distinct but superimposed on one another in some number of dimensions, you need to introduce a new dimension to describe the new coordinate. This is why time is a dimension. Suppose you have a ping-pong ball located at some point in space. Someone then moves the red ball and places a blue in the same spatial coordinates where the red ball once was. Both the red and blue balls can be described as occupying the same point in 3d, so you need a fourth dimension (time) to accurately describe the system.
It's normal to find dimensions confusing. Remember that two of the major mathematical revolutions of the last 150 years (topology and differential geometry) as well as a major branch of physics (relativity) were built around questions exactly like those raised in this thread.
