If full employment real interest rates are determined by investment (the demand for loanable funds) and savings (supply of loanable funds), is there any reason to suppose that there is a lower bound on real interest rates?

In theory, the loanable funds market can reach an equilibrium at any level and thus there should be no lower bound on real interest rates. Nonetheless, in practice, people only encounter nominal interest rates in the real world and it is reasonable to believe that there is a lower bound on nominal rates near zero. This is because you would not put your money in the bank if you could get a better return from simply having your cash under your pillow. This means that nominal interest rates cannot fall appreciably below zero putting a constraint a real interest rates since:

real interest rate = nominal rate - inflation

However, inflation would seem to negate our constraint on real rates since inflation could be very high allowing real rates to reach very negative levels. Thus we appear to be back where we started but we must analyse why nominal interest rates would be so low?

The simple answer is central banks have attempted to revive Western economies since the financial crisis of 2008 implying that performance has been weak. The crisis destroyed confidence meaning firms put off investment and consumers slashed spending acting to reduce the demand but increase the supply of loanable funds sending the full employment or equilibrium real interest rate spiralling downwards becoming negative. However, owing to the low levels of demand in the economy inflation remained very low and since central banks reduced rates to nearly zero (0.5% in UK), there is a floor on real interest rates near the zero lower bound since:

real interest rate = nominal rate - inflation

r_min = 0.5 - 0.7 = -0.2

0.7% was the average inflation in UK between 2014-16 thus there was a floor on real interest rates of -0.2%. This problem is called the zero lower bound and has serious consequences as highlighted by the theory of secular stagnation. We can discuss the consequences in depth and even causes/solutions.