In Pearson Distribution System's range for $x$ For some Type of the Pearson distribution system, the $x$ is bounded for a certain range. 
Now if I want to get the probability of say certain $x$ value and yet it is out of the defined bounds of a specific Type of the Pearson system, does it mean that I can't use the Pearson Distribution system for this situation? (Because it only gives the value equal to zero.) 
However, when I check using the Monte Carlo simulation, the probability is not equal to zero. 
Is this a limitation of the Pearson distribution system?
 A: The Pearson system is a location-scale family. 
For example, consider the Pearson type I, which is a four parameter beta - while the standard beta family is on (0,1), the corresponding distribtuion in the Pearson system is on any finite interval you like.
You should be able to find suitable estimates for the boundary parameters (avoiding impossible values for estimates -- i.e.  estimates that don't leave data hanging outside the bounds). This may contradict matching the mean and variance, however -- method of moments type estimation won't necessarily pick values that do this.
Similar comments apply to the Pearson types that are bounded on one side.
If you're fitting the distributions to your data, you should be able to avoid this problem but it may involve considerably more effort than the usual method of moments -- and even directly attempting to maximize the likelihood without some form of regularization / modification may lead to problems (see the comments below).
So it's not that you can't use the Pearson system with your data -- but you might have to modify the way in which you're using it. Alternatively, you might consider using another system, such as the Johnson distributions.

Note on maximum likelihood with the four-parameter beta
[Some similar issues can come up with the three parameter gamma as well -- this is Pearson type III.]
There can be some issues with maximizing likelihood for the endpoints in the four parameter beta case, particularly in non-large sample sizes or for some values of the shape parameters. See for example Carnahan (1989) [1].
Wang (2005) [2] discusses a Bayesian approach, using a particular form of prior to regularize the problem. 
McGarvey et al (2002) [3], discuss the poor performance even of Carnahan's suggested approach in small samples (n<80) and propose a heuristic they say has better performance.
[1] Carnahan J.V. (1989),
"Maximum Likelihood Estimation for the 4-Parameter Beta Distribution",  Communications in Statistics - Simulation and Computation, 18(2), 513-536
[2] Wang J. Z. (2005),
"A Note on Estimation in the Four-Parameter Beta Distribution",
Communications in Statistics - Simulation and Computation, 34(3), 495-501
[3] McGarvey, del Castillo, Cavalier & Lehtihet (2002),
"Four-parameter beta distribution estimation and skewness test,"
Quality and Reliability Engineering International, 18, 395–402
