I just tried it and it works fine:

50.0 +- 50.0:  43.1
50.0 +- 50.0:  18.8
50.0 +- 50.0:  38.2

50.0 +- 50.0:  32.8
50.0 +- 50.0:  13.4
50.0 +- 50.0:  53.8

50.0 +- 50.0:  67.6
50.0 +- 50.0:  25.2
50.0 +- 50.0:   7.2

50.0 +- 50.0:  76.4
50.0 +- 50.0:  10.9
50.0 +- 50.0:  12.7

50.0 +- 50.0:  89.9
50.0 +- 50.0:   3.1
50.0 +- 50.0:   7.0
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There is risk of getting trapped and not converging on an answer. E.g. if the first number is 99, then the second number will still be randomly generated between 0 and 100 and will be thrown out repeatedly unless it's less than 1. Ditto for the third number but in an even tighter range.

However, I think that can be avoided by constraining each random number to be in the range from (mid - sd) to min(mid + sd, remainder). In the example above, the second number would be chosen between 0 and 1. Assuming that was 0.6, then the third number would be chosen between 0 and 0.4. That would be guaranteed to converge quickly.