You certainly can make a forum that saves all its information in flat text files. However, there are a bunch of problems with that. First of all, it's much slower than a database if you have large files. Databases will cache queries, which makes them much faster. It can also be difficult to update information in a text file; you have to search through each line for the value you want.
The only use of such a system is for hosts that provide PHP or another server side language but not databases. These hosts are few and far between, so you won't have a large user base. However, it can definitely be a learning experience for you. Go for it!
@Faran--there are no security problems with using flat files

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@Floobynooby--he doesn't need a database.