Does the outcome of aromatic nitrations depend on the concentrations of nitric acid and sulfuric acid?
Nitration occurs as electrophilic aromatic substitution and activity of substrate varies depending on exact structure of ring system and substitutes in the ring. In general electron acceptors (nitro-, acyl-, nitrile- and sulfogroups) deactivates the reaction, while electron donors (hydroxy-, aryl-, alkenyl- and to lesser extent alkylgroups) activates the reaction. As nitration occurs step by step, it will advance to dinitroarene only if mononitroarene is not deactivated too much by introduced nitrogroup, and same for other nitroarenes. However, different nitration agents have different nitration activity with 100% nitric acid in oleum being one of the strongest, usual nitration mixture (70% nitric acid + 98 % sulfuric acid) being strong, 70% nitric acid being moderate, diluted nitric acid being gentle and sodium nitrite in acidic solution being one of the gentlest. The picture is further complicated because strong nitration agents are also strong oxidizers, capable of breaking apart delicate molecules like phenols. Given that, nitration is often performed in several steps, with first 'gentle' nitration to introduce the first nitrogroup without breaking the molecule and then further nitration with stronger agents (see, for example, synthesis of styphnic acid). 

Exhaustive review of the topic is beyond what a simple post can cover. For further insight into the reaction I recommend to consult a decent university-grade organic chemistry textbook, like 'March's Advanced Organic Chemistry'.