Can I add streptavidin (with PE, FITC, APC etc.) to my biotin-conjugated antibody before incubation with my cells? I want use this antibody to FACS analysis.
Since the affinity of streptavidin and/or avidin for biotin is among the strongest non-covalent attractions known in all of biochemistry, I think you will be fine in doing so as long as you do not alter drastically the ionic strength of the down-stream solutions you use for your assay. Assuming your protocol is all at physiological pH - I would say yes, this should work quite well.
I believe the general idea of adding the streptavidin-bound marker after antibody incubation is to prevent the resulting streptavidin-biotin complex to interfere with the antibody binding to your cells.
I would rather not do the experiment in the order you are planning. After chemical labelling each antibody is likely to carry multiple biotin residues. Streptavidin can bind up to four molecules of biotin. This leads to the formation of high-molecular weight complexes of antibody and streptavidin which might precipitate (become insoluble). The better option is the classical approach: Bind antibody, wash away unbound antibody, then add streptavidin.
As much i'm concerned , ultimate purpose of Streptavidin adding later on is -- after biotin binding with primary amines of cells, streptavidin will bind with biotin and cells will settle. If you bind streptavidin - biotin first, then streptavidin might will occupy all biotin cell binding groups and you might will not find streptavidin-biotin - fluorochrome on FACS analysis.
So, my answer is - you can go for a trial but better to follow later on incubation with streptavidin.