This has been discussed before in detail. Short answer is it depends, could end up being better, worse, or not noticably different. Correctly done it can be equal or better (i.e. using 24 bit output on 16 bit normalized data gives you some headroom without loss of fidelity). Poorly done and you can be listening to 8 bit audio (FYI, it is rarely the former). So yes, it doesn't hurt bit-rate, it can hurt bit-depth.
You can think of this concept quite simply with image processing for example. Lowering brightness on a raw picture only lowers the amplitude of each pixel. Assuming we have a 8-bit grayscale in a 16 or 24-bit data format, there are brightness adjustments we can apply that will not throw away any actual contrast. But if there is a 16-bit grayscale image utilizing 16-bit of contrast and your result file is 16-bit, than any brightness change (adding or subtracting amplitude), will result in loss (meaning your whitest whites don't get any whiter or your darkest darks won't get any darker).
Analog adjustments using the same analogy, would be manipulation of lighting or optics with lens and lamps, etc. Again the end result still depends on the quality of said analog components.
Now throw in a 'noise-floor' factor into the mix with the preamp vs soundcard output, and basically the answer will still be it depends. In general, significantly attenuating your music digitally only to significantly amplify it is a bad idea. And often this is exactly what is the situation since you are using your computer to attenuate only to amplify that result.
Even ignoring sheer loss of bit-depth and thinking about just the analog realm, it can be in many instances a bad idea. You may have 24-bit digital headroom to play with, but the least significant bits might not resolve so well anyways by the time it comes out of your soundcard. So playing your 16-bit audio normalized and set towards the most significant bit (i.e. louder) is still usually better than playing that exact same 16-bit data towards the least significant bit. True there is no bit-depth loss, however I can guarantee you that your soundcard does *not* resolve full 24-bit quality such that your least significant bits are reproduced as well as your MSB's anyhow. The *true* noise floor of 24-bit audio would be -144 db or so, and no audio circuit much less soundcard I know of approaches that. That means all the bits you push towards the LSB are masked by noise floor anyhow.
Going back to the image processing example, this would be like applying a darkening on an image on an LCD monitor without truly black blacks. Maybe there isn't any digital truncation, but your analog component's performance can still mask it.