Yes - you can open multiple reports in Monarch (up to 1024). To do this, open one report, open your model, navigate to the Report window, go to File, Open Report, browse to the right folder, select the reports you want. You'll be prompted to retain the open report and model - tick yes to both options.
When you have lots of reports open, you might find a calculated field defined as rsplit(File();2;"\";1) is helpful - this will let you know what filename each detail line was found in.
Just to be crystal clear, Olly's advice is perfectly sound if each of the text files that you receive is an instance of the same report, say P&Ls run for different departments, so that every file has the same basic structure. In that case, it's a very easy solution.
But if you're receiving a collection of different report formats, then automating that sort of manual processing of the Monarch export process is possible in a couple of ways, such as batch files using Monarch's command line parameters, or by using Monarch COM interface that allows you control Monarch via custom programming, such as can be created with Excel's VBA.