You can indeed have more than one license in place, at the same time, and either give the user of the code some freedom to pick the license he likes best (see dual- or multi-licensing on Wikipedia), or require him to obey the terms of both licenses, as long as they do not contain contradicting requirements.
A popular model is to have one license for a "work as a whole" (containing many source files), and different licenses for individual source code files, as long as the license for the individual source file is more permissive than the license for the work as a whole. For example, the GPL is a license where the text explicitly mentions it applies to the "work as a whole". That is why the Linux kernel is under GPL, but some kernel modules contain files which are under MIT or a modified BSD license. So as long as you use or publish those files individually, and not as part of the Linux kernel, you only have to follow the terms of the MIT or BSD license.
In your specific case, I would expect the MIT license has to be applied to the "work as a whole", and the combined MIT & Apache 2.0 for the files which contain the Apache license. The latter one is - at least slightly - less permissive than the MIT license. For example, the Apache license requires you to mention changes to the original code. IANAL, but such a term does not seem to contradict the MIT license. For example, when you change the file with the Apache license embedded, I would expect you have to mention those changes. When you change a different file, I would expect you are not required to do so. There are more differences between the two like a patent clause in the Apache license which is not part of the MIT license. So what you will actually have to check is if your intended usage of the code confirms to both licenses. And if you are not sure, ask a lawyer.