It may sound like a strange question in light of all the talk about "too big to fail" during the last few years. But, believe or not, the idea that bank concentration has an impact on real economic activity isn't the standard view. Here's from a recent blog post by NY Fed economists Mary Amiti and David Weinstein:

The notion that financial institutions are large relative to the size of economies is not something that plays a prominent role in traditional economic theory. Macroeconomic textbooks tend to treat economies as composed of representative firms that are infinitesimal in size compared to any given market. As a result, positive and negative idiosyncratic shocks [movement in bank loan supply net of borrower characteristics and general credit conditions] to financial institutions cancel out due to the law of large numbers. 

However, this representation stands in stark contrast with the reality of concentration in financial markets. A striking regularity is that a few banks account for a substantial share of an economy’s loans.

Starting from this basis, Amiti and Weinstein have examined Japanese aggregate bank lending data and other aggregates and were able to demonstrate the following: banks matter, bank concentration matters, bank lending matters. No small feat.

On the issue of bank concentration and aggregate lending, they found that

...if markets are dominated by a few financial institutions, cuts in lending due to some change in financial conditions in just a small number of banks have the potential to substantially affect aggregate lending. Moreover, if firms find it hard to find good substitutes for loans like issuing equity or debt, then it is possible for their investment rates to fall as well. 

As for their take on banks' impact on the real economy, the conclusion to their paper (on which their blog post in based) gives a good summary:

Our paper contributes to this literature by providing the first evidence that shocks to the supply of credit affect firm investment rates. We find that even after controlling for firm credit shocks, loan supply shocks are a significant determinant of firm-level investment of loan-dependent firms. This result is particularly surprising because our sample is comprised of listed companies that have, by definition, access to equity markets. Moreover, the fact that so much lending is intermediated through a few financial institutions means that idiosyncratic shocks hitting large financial institutions can move aggregate lending and investment. We show that about 40 percent of the movement in these variables can be attributed to these granular bank shocks. This means that the idiosyncratic fates of large financial institutions are an important determinant of investment and real economic activity.

And the implication for policy, according to Amiti and Weinstein, is significant. Here is the relevant excerpt of their blog post on this point:

...[P]olicymakers without detailed information on the major financial institutions are likely to have a difficult time understanding the causes of lending and investment fluctuations. A large portion of Japan’s aggregate economic fluctuations can be traced to the country’s banking problems. 

While many researchers have focused on the implications of banks being “too big to fail,” we show that even if large banks do not fail, granular bank shocks can have substantial impacts on aggregate investment. 

For example, reductions in bank capital at large financial institutions can cause investment declines by firms that would like to borrow, while recapitalization of the right institutions can stimulate investment. In sum, this study shows that what happens to large financial institutions is important for understanding aggregate investment behavior. 

While their paper looks specifically at Japanese data, the authors suggest that the overall conclusions are relevant to the situation in the US given that it too has a very concentrated banking sector.

Amiti, Mary and David Weinstein, How much do banks shocks affect investment: Evidence from matched bank-firm loan data, NY Fed staff paper 604, March 2013