Do all cells produce the same proteins?
Each cell will indeed have the same DNA sequences and ability to produce any given protein. However, there are certain factors (transcription factors) and cellular conditions within a cell that dictate which proteins are produced. If the conditions are right then only certain proteins will be produced depending on what type of cell it is. This process by which a cell continues to express selected proteins and becomes specialized for a specific function, is called differentiation. Expression of proteins that are not required can be prevented by transcriptional repressors (or by epigenetic mechanisms such as methylation of DNA sequences). So to answer your question, while cells contain the instructions (DNA sequences) to produce potentially any protein, not every type of protein will always be produced. 

Keep in mind that my answer is very basic; the answer isn't as simplistic as I'm making it out to be.