This is a common problem with these radios. I also recently purchased a 328i (a 1996), and it also had the dim radio display problem. I happen to be an electrical engineer, so I figured I could solve the problem myself. As it turns out, I
was able to solve the problem, but it wasn't as simple as I expected it to be.
The buttons of the radio are illuminated by surface-mount LEDs, but the LCD display and the tape deck opening are illuminated by small, 12-volt, incandescent lamps. The Radio Shack equivalent is
#272-1092. It is possible that one or more of these bulbs have burned out, but what happens more often (and what had happened in my case) is that the bulbs themselves are fine, but there is a problem with the circuitry driving them. In particular, it is common for one or more of the solder joints that anchor some of the components of the lighting circuit to the circuit board to crack and fail. The symptoms you describe are consistent with this, since the cracks are typically small enough that sometimes the vibration of the car can close the cracks, leading to the lights working intermittently.
Arturo Gossage wrote
an article on the subject in 2002 for
Boston Bimmer.
If you have a basic understanding of electronics and some skill with a soldering iron, you can probably fix the problem yourself. It helps if you have access to a 12-volt power supply (so you can plug the radio up and test it outside the car). You'll need to remove the top cover and cassette assembly from the radio, so you can easily access the mainboard. The component that seems to most commonly develop a broken solder joint is a surface-mount resistor located on the mainboard, towards the front of the radio, to the right of connectors where the front panel connects to the mainboard.
A magnified view of this resistor is pictured in Arturo's article.
One way to find the problem solder joint is to connect the radio to a 12-volt source and then probe around the mainboard, pressing on each of the components. When you find the one with the broken solder joint that is responsible for your display not lighting up, the display should light up when you press on it (because by pressing on it, you've closed the crack in the solder, re-making the electrical connection). When you find it, simply repair the solder joint, and you're back in business.
If you need any help, just let me know. Or, if you'd rather not tackle this yourself, I'd probably be willing to do it for you ... but I have a full-time job and a family life (and it takes some time to do the repair), so I unfortunately can't do it for free. I'm not looking to get rich or anything, though - just some small compensation for my time. If you're interested, send me a PM, and I'm sure we can work something out.
Alternatively, there are a few businesses that specialize in factory radio repair (do an Internet search for
BMW radio repair), or you might be able to find a replacement radio on eBay for cheap (you have the C33 model, BMW part number 65 12 8 364 944). You also might consider upgrading to the CD43 radio (BMW part number 65 12 6 909 884). This radio was available for a short time on the Z3, it has an in-dash CD player rather than a tape deck, and it's a perfect match to the 328i interior. It also has
better sound quality than the tape-deck radio. This is available from
Bavarian Autosport for $730, but you can probably find it much cheaper on eBay. (I just bought one of these myself a few days ago.)