When I asked Google for help, it led me to a page talking about how to use SyntaxHighlighter with Blogger. The post itself didn't contain any new information for me, but the comments were useful. "Robje" had apparently experienced exactly the same problem as me. "Debiprasad" then asked, "What is your settings for 'Convert line breaks' set to? In my case, if I set it to 'Yes', it's inserting <br/> into codes defined under <pre> tag." Ah ha! I looked on the "Formatting" subtab under the "Settings" tab in the Blogger control panel and discovered that mine was set to "Yes". Ack! What an annoying default! With it set to "No", it seems to fix the problem.
In case that doesn't work for you, Google also brought me to a page on what must been the old home for SyntaxHighlighter. Once again, the gold was in the comments. "fahd.shariff" mentioned that if you use the post editor in Blogger in Draft, this isn't a problem. As I said earlier, I gave this a try before finding the preferable solution and it worked for me so it's a reasonable solution to fall back on.