For anybody that reads this in the future, the issue is that I was being given a cell with a huge matrix crammed into it. So, using Tarunbir's answer, all I had to change was to use SigPlusNoise{1} in combination with the (i, :, j).
My code is now listed below and runs fine.
for i = 1:1000
for j = 1:10
New_Sig = SigPlusNoise{1}(i,:,j); %Drawing signal from given matrix
SigIn = New_Sig; %Establihing input signal
cheby = LPF_ChebyI_V1(SigIn,Fsamp,Order,BW,Rp,ImpPlot,PrntPlots); %Filter function
New_data(i,:,j) = (cheby); %Making new matrix
end
end