No, at least not directly.  As has been alluded to by others, a bank run is a sign of rapid deflation.  A bank run is caused by the sudden loss of faith in the soundness of an institution.  A national bank run would indicate the loss of faith in the soundness of the Federal Reserve System & the FDIC.  The result of large numbers of people trying to get their money in cash is a sudden and drastic increase in the demand for currency.  The law of supply and demand, of course, leads us to the conclusion that the value of cash would rise to close the supply gap; which would be large because the FDIC has the option of taking up to 60 days to follow through on the insurance claims without needing outside approval of a longer time period.  There are good reasons why this grace period exists, but the result is that there would be roughly a 60 day period wherein the supply of currency in the economy is significantly short of the demand; and that is assuming that the public actually trusts that the FDIC can follow through, which is mathmaticly unlikely in any national event at the present time.  Currently, the FDIC is insolvent due to so many banks going under the last two years.  The net result of bank failures in the longer term is also deflation, as the failure of a bank is usually related to too many of it's own loans going sour and never getting repaid.  Debts forgiven in bankruptcy court are equivialent to currency, previously willed into existance by the power of fractional reserve banking at the inception of the loan as credit, ceasing to exist.  On a large, national scale, the net result of such events is the contraction of the currency base as it's perceived by the economy at large; (the business owner views credit as being functionally equivialent to currency) and the denial or loss of credit availability results in the academic definition of deflation, i.e. the reduction of currency in circulation relative to the mean economic activity that it represents.  Changes in consumer prices are affected by too many other variables to point at any group of prices at any particular time and determine if inflation or deflation has occured.   A sharp reduction in 'velocity' can have similar effects to reductions in gross volume, and that is usually what we call a "recession".

Of course, a national bank run could easily result in a political 'intervention', resulting in the political forces triggering a hyperinflation; but a national bank run is mathmaticly deflationary.