Whether to publish one big paper or many smaller papers for a given research project? Let's say I invented a system to solve a problem. To run this system, I made my own algorithm.I also created some other things for that system. The main contribution was supposed to be the system.So is it a good idea to have as many research papers out of the project as I can? Or to publish a single research paper?

I have seen a lot of researchers where they were targeting a single problem and they proposed a single solution. Now, what they would do is write a research paper for every component separately. Then they would write a single research paper showing how all the components would fit together.

So is it a good idea to try and increase the number of publications you can have out of a single research project? 

My own personal opinion is that the quality of your research matters not the quantity. But I have also seen a number of institutions requiring a specific number of publications to even apply for their jobs.
The way I see it, there are a number of factors at play: 


Your goals: Do you want your paper to be published in a high-impact journal? If so they will most likely be interested in the whole story rather than a small piece of it. 
Readability/General appeal: Can you make a coherent story with individual components? Will they all be interesting for wider audience on their own? In other words, if you opt for multiple papers out of one project; can you make sure these will be able to stand on their own? I personally think going for multiple papers is only valid when combining them into a single big paper would push aside some of your interesting results (or methods) to a metaphorical corner
Limitations imposed by the target journal: Can you actually put together all that in a single manuscript? In biomedical research you always get a limit on the number of words in the manuscript, and there is only so much you can put in supplementary. 


With regards to quality-vs-quantity, I heard that early on in your career quantity is more important while as you become more and more senior, quality becomes the main concern. I am often told that as a PhD student I can, and should try to get involved in as many papers as I can. Around the time I do post-doc, however, it's time I pay a lot of attention to where I put my name and try to work on a good paper, preferably in a high-impact journal.