First thought was no, but then I started playing around. I haven't gotten it to work perfect, and part of that is that I'm not positive of your requirement.
The plan would be to divide 10 by a RandBetween formula. You'd need to take into account the fact that each cell can only be big enough to allow at least a 1 in the remaining cells. So your first random cell is actually =RandBetween(1,10-4). You'd have to reduce the upper limit of the random statement based on the value returned so far to allow at least 1 in the remaining cells.
The other part I'm not convinced keeps 100% randomness is the fact that by always starting with the first cell and then reducing the values available each time your later cells have a smaller chance of being large. (They could only be large if the first 4 cells randomly were 1, for example.)
To get around this, you could have 5 rows where the formulas are adjusted so that each cell gets a chance of being first, second, third, etc. You can hide these rows and have one row at the top with your real answer. That one would reference one of the rows randomly using a vlookup based on RandBetween(1,5) formula.
Confusing enough? Because I'm barely following it at this point, but I know it would be possible. Worth it? Who knows, but possible it is.