One can always expect that the estimated coefficient on a given independent variable, call it X1, will change with each additional independent variable (X2, X3, etc.) added to the linear regression because:

When new independent variables are added to a linear regression, the estimated coefficient on the existing independent variable X1 will change because the new variables could capture some of the unique explanatory power that X1 had on the dependent variable. This contributes to the coefficient on X1 changing - it can become higher, lower, or stay the same. The changes in the coefficient of X1 are dependent on the amount of explanatory power of the new variables introduced, and the correlation that exists between X1, the new variables, and the dependent variable.