Can I mix and match different brands of cleansers and moisturizers?

It's okay as long as you read the fine print. It's far too easy to go overboard with strong ingredients if you're not careful. For example, you may be using a cleanser that contains a salicylic acid, then applying a lotion with some kind of alpha-hydroxy acid (AHA), and not even know it. This kind of overlap and excess can unintentionally cause serious irritation. You must look at the ingredients carefully and then evaluate whether the new product is compatible with your existing regimen or giving you double doses of an active ingredient. (This is important to consider when you receive a free sample at a department store too. Find out what's actually in it before you put it on your face.) For this reason, using a skin care system is a rational way to go. A line of products that is made to work together harmoniously is simple to use and will prevent the possibility of exceeding the limit on harsh substances. For someone with sensitivity, this is especially important, because you don't want to keep introducing new ingredients and chemicals to your skin.
Additionally, switching often doesn't give a product with active ingredients a chance to work. Many cosmeceutical ingredients (such as Retin-A or retinols) can take six weeks to take effect. Just because you don't see a noticeable effect overnight, that isn't a reason to try something new. The main reason to change a product should be if it's causing redness, itchiness, pain, or irritation - not just boredom. Many people get impatient because they don't have realistic expectations for their products, and they switch them for another and expose themselves to a new group of chemicals, setting themselves up for problems.