Advice on choosing a really fast DBMS

I am working on a project that does real-time monitoring of data being transmitted, and logs if there are any of the various type of errors, etc. The CRUD operations are real-time. We initially planned on using PostgreSQL, but the problem we are facing is that PostgreSQL is not fast enough to handle real-time CRUD operations even after tweaking it a little; same goes for MySQL and other biggies. SQLite performs a lot faster than them, but is soon almost dead when the database size reaches a few hundred MBs. Another constraint is that the monitoring is to be done over network.

Is there any database that can handle such fast operations? Or should I opt for a NoSQL database?

EDIT (Regarding design):

The design is normalized to the maximum extent it was possible. The stored data is almost mutually independent, so there are very few joins. Also, I said "a few hundred MBs" just as a reference. The actual databases that we work upon are multiple GBs in size. Lots and lots of data is inserted every second and retrieved also.

Talking about PostgreSQL, it takes 5-7 times the time taken by SQLite in the tests that I ran on my data.

EDIT (Regarding speed):

I'll like to mention a worst-case scenario that can happen.

Suppose that the main application is being used at 10 instances (or PCs). They all interact with a cental DBMS and insert data into it. Now, every single app will have many threads doing some operations on the data that is being transmitted in real-time. The app reports if there is erraneous data in the streaming or not. And since the data is analyzed at packet level, lots of errors can happen in a sec. Based on some very basic calculations, the worst case may require an insertion rate of ~3k rows per second per instance, with each row having some 8-10 associated columns. I tested such a test on my machine(4GB ram, QuadCore), and SQLite is able to do this in ~1 sec over network. I had tweaked PostgreSQL a little, and it did that same in ~5 sec (I'd admit that I did not optimize it a lot, as I am not a pro in the DBMS field). But the bottleneck comes as the db size grows bigger in SQLite; the insertions seemingly go on almost fine, but the reads take a lot of time. I had tested it with a DB size of 3gb myself.

Our major concern is the insertions, which, in worst case will be ~3k per app instance and in average case ~500-1k insertions per instance.