Ok. Thank you for the data. The plot does help a lot actually. At first glance, the plot of the fit itself looks quite reasonable. But then I focused my old bleary eyes on it, and I see the sinusoidal oscillations in the curve near the flat regions. (When I said I might guess what the problem was, this is what I was thinking.)
That is a common consequence of trying to fit a flat curve (or a curve with flat or nearly flat sections) with a sum of sine waves. Higher frequency sine functions will just introduce more (higher frequency) wiggles in the flats. Take enough terms and eventually the bumps go away, at least to the point you do not see them. But this just argues you did not take enough terms in the model.
It is exactly like the presence of Gibbs phenomena in a fit. In fact, if you look at the figures in the wiki page I link here, you will see that same behavior.
Again, taking more terms in that series will improve the fit, beause the oscillations will eventually cancel out. You don't have any sharp transition in your data, so there are no large overshoots near the edges. But even so, there will still be the same characteristic oscillations in the region of the curve that want to be nearly flat.
So, how can you improve on that result? Again, the simple solution is to just throw more terms at the problem. If you insist on having a model that is a sum of sines (and cosines usually) that will work, to drive down the oscillations. They will still be there of course, only at a finer level. Nothing you can do will prevent that in such a model.
Alternatively, you could use a spline model of sorts. Perhaps a smoothing spline would be a good choice, or a least squares spline. But splines have their own issues of course. People always want to write down the function, and you simply cannot write down some nice looking function from a spline. You can use them to evaluate, or make a nice smooth plot.
