When I overexpressed a gene, it caused the cell growth to be severely affected and a large cell phenotype was observed. I am wondering is this normally happens when a protein is in excess? Or this phenotype is specific to certain type of protein?
I would guess that this is a phenotype specific to the protein you are expressing, the cell type you are using, or a combination of the two. I have never observed a noticeable increase in cell size resulting from protein overexpression. If you haven't already, maybe test whether the phenotype is reproduced when a different (unrelated) protein is overexpressed in the same context under the same conditions.
Exactly: it depends entirely upon the gene you chose. If you choose an oncogene, for instance, you might well see a change in growth rate. On the other hand, if you over-express something like actin (or some other growth-unrelated gene), you would not expect that phenotype.
So you just have to ask: is your gene reasonably likely to be related to growth phenotypes? If so, then your results are expected. If not, either you may have learned something new, or you may need to conduct additional controls to ensure nothing is wrong with your protocol.