My background: I'm a senior staff integrated circuit designer for a major high-performance analog manufacturer. I have a bachelor's in computer engineering and am working on a master's in electrical engineering.
The bottom line is that computer engineering and electrical engineering are essentially the same profession. They're both heavily hardware-oriented. About 90% of the material overlaps. In the last year, EEs take a few more classes in the direction of fields, power electronics, and RF, while CpEs take a few more classes in the direction of programming, operating systems, etc.
With a little experience, most EEs can do "computer engineering," and most CpEs can do "electrical engineering." I chose CpE because I felt my programming skills were already strong, I wanted a lot of exposure to hardware, and I didn't intend on doing any work with power electronics or antennas.
You won't be prepared for a genuine "design" position with any kind of bachelor's degree, really -- particularly not designing microprocessors, which is one of the most highly-specialized areas of electronic design. With a bachelor's and a few years of work experience, you can easily get into ASIC design, though.
- Warren