If the desire is a bounded, symmetric, continuous distribution that approximates a normal, consider the beta with, eta,gamma equal. The pdf is then bounded between [0, 1] with mean gamm/(eta+gamma) --> gamma/(2*gamma) --> 0.5 for eta==gamma.
As for the normal, you can shift and scale the generated RNVs generated from random by whatever is needed to match the target range.
The 'pdf' normalization inside histogram results in the red overlaid normal; scaling the N() pdf to match the peak bin in the histogram results in the black overlay which emphasizes the extra weight of the beta towards the central tendency as compared to a normal. But, you can produce a bounded random variate this way that with the very nebulous requirements for the underlying error distribution could surely serve the purpose.
The above was generated by
where the magic constants were obtained by getting the maxima of the histogram binned values and the pdf peak
The above uses functions in the Statistics Toolbox...