Problem with nlinfit parameter estimation

I have a model with 9 differential equations defining dynamics of 9 different variables. I have total of 19 parameters to estimate from data fitting. But I have data only for 3 variables at 4 different times i.e. total of 12 data-points to fit. First I tried manually to fit the data and have a rough estimate of parameters. But when I use the nlinfit with these estimates as initial guesses, following messages appears several times:

Warning: Matrix is singular, close to singular or badly scaled. Results may be inaccurate. RCOND = NaN.

Warning: Failure at t=2.130939e-010. Unable to meet integration tolerances without reducing the step size below the smallest value allowed (4.135903e-025) at time t.

And after some 10-15 iterations the process stops saying the system is overparameterized and the parameters cannot be estimated. I think the system is too sensitive to parameters and the ODE solver is not able to solve it after some changes in initial guesses. I have used all the ODE solvers (ODE 45,23,113,15s and 23s) but the result is the same.

What should I do? Is there any other method for this problem? Thanks in advance!

[Star Strider]

You simply cannot uniquely identify 19 parameters with 12 data.