Actually, you can use all three versions of Visual Studio at the same time. I've got one of my machines set up this way, and it works great if you need to work with projects from each of the versions. If you don't need Visual Studio .NET 2002, just uninstall it before you install Visual Studio .NET 2003.
As far as compiling your Visual Studio .NET 2002 projects with Visual Studio .NET 2003, you shouldn't have much trouble. Depending on what you're doing in the .NET code, you'll probably see warnings about depreciated interfaces starting with .NET Framework 1.1. It shouldn't be that big a deal. Since it sounds like you're staying with VC6 for the C++ portions, you should be fine. If you were going to move those portions over to Visual Studio .NET 2002/2003, you'll find that the new compiler is much tighter on what it allows, so you'd end up tweaking a lot of code to get compiles to work. I'd bet serious money that making your code work with the tighter compiler will fix a couple of bugs too.
Probably the worst issue Visual Studio .NET 2003 is that once you open a project in it, that project can't be open any more in Visual Studio .NET 2002 because the project and solution file formats changed.
This was first published in November 2003