Inter-filter bit width
A source of dynamic range issues can be quantization leading to noise increase in your summation if in the datapath or errors in the frequency response if the coefficients themselves have too much rounding error. Quantization noise for an $K$-tap FIR filter can increase as much as $10 \log_{10}K$ in dB when you scale the signal at the input of the filter.  This is the reason for extended precision accumulators; good practice is to let the filter grow the signal and then attenuate after the filter.  Also equally important is that your coefficients should be (as a rule-of-thumb) at least $DR/6 + 2$ bits in precision where $DR$ is the stop-band rejection requirement of your filter.

To the extent that each signal at each tap in your filter can be considered uncorrelated (and this holds if the signal is white, however your first filter will provide some correlation between delay taps reducing the total increase in the noise). 

See the figures pasted below that compare the three ways to scale the signal.

To explain these figures, this is the case of a filter with a fixed precision datapath, and a case where due to the input level of the signal, the filter would overflow (since the signal within the bandwidth will increase by a factor of $K$ in magnitude). To keep the filter from overflowing, the filter designer can scale the input, or scale the coefficients, or (the right answer) is allow the filter to grow in precision, and then scale the output.  In the first two cases, the noise would be the quantization noise due to the scaling, and grow in magnitude by the $\sqrt{K}$ (stated differently, variance grows as $K$, standard deviation grows by $\sqrt{K}$), in the last case the noise is just the quantization noise due to the scaling (6.02 dB/bit + 1.76dB for a full scale sine wave) without further growth.  

(Note $\Delta^2/12$ denotes the variance due to quantization noise, where $\Delta$ is one LSB step size)