In Windows 10 , I notice saving a lot of pictures really slows things down in Photoshop. I like to save pictures to post on Facebook and by the end of the month I usually have about one to two thousand saved and things run slow. At the end of each month I load & save all the pictures to a DVD disk and delete them off my desktop PC. It than speeds back up until I download more.
This may be less the case these days, with how cheap memory is, but it used to be that computers would use a percentage of your hard drive as a "scratch disk," expanding the working memory. The more memory available, the faster the computer could handle multiple tasks. Hard drives were slower than the chips on memory cards, but it was better than not having the expanded capacity.
If you're low enough on physical memory (space on your hard drive), it could be eating into that scratch disk, which can slow your computer down. Likewise, if your system has too many programs running at the same time, it could be eating your system memory, which can have a big effect on performance.
Like the pipes in a house, they'll get clogged over time. We all accumulate lots of programs over time, it's good to do a purge now and then and just keep the ones you really use. Also keep your files organized, and if defragging the hard drive is still a thing, schedule that once every few weeks.
The short answer is yes. The long answer is similar to Nice Gaijin's answer. They do have something like a "scratch disk" currently. It is called a page file, it will use the hard drive of your computer as ram when your current ram is currently all in use. You can increase or decrease the size of this file but they better fix is to either install more ram or delete things from your hard drive that you no longer use.
I'd like to note that paging has nothing to do with how much hard drive space you have available (at least not directly; see below), but rather has to do with how much RAM you have. Very different. When you run out of RAM, the system has to store excess RAM somewhere to prevent the whole system from crashing, and typically the only choice is the hard drive, which is slow. Hence, paging (or swapping, as it's called in Linux). If this is happening, you're pretty much certain to be running slow no matter what, so one possible solution is to upgrade your RAM. Alternatively, you can try to reduce your RAM usage.
Hard drive space can technically affect speed, but the mechanism is unrelated to RAM. Rather, what can theoretically happen is as you lose hard drive space, your data can become more fragmented on the physical hard drive, which can make access times longer. It's like if you have a full notebook, so you add new notes to scattered blank spaces in pages throughout; doing so means you have to look at more pages and flip through more to find the right ones, which makes you take longer to read the notes.
That being said, I said "theoretically" because this is unlikely in a modern system. Modern filesystems like ext4 and NTFS are designed to minimize fragmentation in the background.
"Too many programs installed" shouldn't affect speed, but too many programs running at one time certainly can. This is more or less because programs have to take turns using shared resources, such as the CPU and the hard drive.
Ultimately, though, there's no easy answer for why a computer might be slow. Fixing a slow system can involve all sorts of interventions: using different software, installing new hardware upgrades, replacing defective components, etc. It all depends on what's going on and what your needs are.
But yeah, check your RAM usage and see if all of your RAM is being used up. If so, that's probably the culprit.
Another thing you might want to check is your cooling system. If the fan on your computer is too dusty, that can make it less effective at keeping the CPU cool, which will force it to work less hard. You can fix that very easily with a little compressed air.