Ordinary kriging is an exact interpolator. Meaning that, if a value is known at a given point, it will predict that same value at that location. (I am not familiar enough with other kriging methods to list which are exact interpolators and which aren't.)
However, it is common to interpolate to populate a raster. In this case, the predictions will be for locations at the center of each cell. It is unlikely that the measured locations are exactly at the center of a cell. In other words: a location you have measured might fall inside a cell without being the center of the cell.
Therefore, when predicting a value for this entire cell, you are predicting the value at a location near the measured location. And the values can differ.
How much the values differ depends on the spatial correlation linked to your semivariogram. The smaller the nugget, the stronger the spatial correlation at a short distance. The stronger the spatial correlation, the closer the values.