Thanks Alexey, but I am looking for more of a loop approach. If I have a matrix of 5 columns and 100 rows with the 5 columns giving the interval values for the cubic spline fit and the 100 rows being the 100 cubic spline fits I would like stored for. I would like a loop that would take the 5x100 row matrix and output the parameters for the 100 fits. I don't know Mathematica that well but attempted to do this and failed. I tried the following:

For[i = 1, i < 101, i++,

F1[t_] = k1 + k2t + k3t^2 + k4*t^3;

F2[t_] = k5 + k6t + k7t^2 + k8*t^3;

F3[t_] = k9 + k10t + k11t^2 + k12*t^3;

F4[t_] = k13 + k14t + k15t^2 + k16*t^3;

F5[t_] = k17 + k18t + k19t^2 + k20*t^3

 Data = Solve[{F1[0] == 0, F1'[5] == F2'[5], F1''[5] == F2''[5],

   F2'[15] == F3'[15], F2''[15] == F3''[15],

   F3'[30] == F4'[30], F3''[30] == F4''[30],

   F4'[45] == F5'[45], F4''[45] == F5''[45],

   F5''[0] == 0, F1''[0] == 0, F1[5] == matrix[i,1], F2[5] == matrix[i,1], 

   F2[15] == matrix[i,2], F3[15] == matrix[i,2], F3[30] == matrix[i,3], 

   F4[30] == matrix[i,3], F4[48] == matrix[i,4], F5[48] == matrix[i,4], 

   F5[88] == matrix[i,5]}]

 {F1[t], F2[t], F3[t], F4[t], F5[t]} /. Data

]

Thanks Gianluca, You have been tremendously helpful. I do have another question for you and the wider community. What if I wanted to try and do a larger number say a hundred cubic spline fits? Is there a way I can pass in all this data and get the coefficients calculated for each of these separately. Basically if I run an experiment and get a bunch of different F1 and F2 values from different conditions that I run the experiments?

Yes, you can do a piecewise definition, or else you can combine two separate plots using Show:

F1[t_] = k1 + k2*t + k3*t^2 + k4*t^3;
F2[t_] = k5 + k6*t + k7*t^2 + k8*t^3;
Data = Solve[{F1[0] == 0, F1[5] == 3, F2[5] == 3, F1'[5] == F2'[5], 
    F1''[5] == F2''[5], F2[15] == 11, F1'[0] == 0, F1''[0] == 0}];
{F1[t], F2[t]} /. Data
Show[Plot[F1[t] /. Data, {t, 0, 5}, PlotStyle -> Red],
 Plot[F2[t] /. Data, {t, 5, 15}, PlotStyle -> Blue], PlotRange -> All]

You can tell Solve what variables you want to solve for:

sol = First@Solve[{F1[0] == Fi1, F1[1] == Fi2, F2[1] == Fi2, 
  F1'[1] == F2'[1]}, {k1, k2, k3, k4, k5, k6, k7, k8}]

You cannot plot the solutions directly, because some constants are still symbolic:

{F1[t], F2[t]} /. sol

You can assign all the remaining parameters the value 1, for example, and then plot the functions:

{F1[t], F2[t]} /. sol /. 
 Thread[{Fi1, Fi2, k1, k2, k3, k4, k5, k6, k7, k8} -> 1]
Plot[%, {t, 0, 2}]

I am not sure I understand your question, but perhaps you will get started with this:

Clear[F1, F2, k1, k2, k3, k4, k5, k6, k7, k8];

F1[t_] = k1 + k2*t + k3*t^2 + k4*t^3;
F2[t_] = k5 + k6*t + k7*t^2 + k8*t^3;

Solve[{F1[0] == Fi1, F1[1] == Fi2, F2[1] == Fi2, F1'[1] == F2'[1]}]