Why can there not be more than one sigma bond in a set of bonds?
Why can there not be more than one sigma bond in a set of bonds?


There can be, even in simple carbon compounds. Bent bonds, tau bonds or banana bonds; whatever you might like to call them were proposed by Linus Pauling; Erich Hückel proposed the alternative $\sigma - \pi$ bonding formalism. Hückel's description is the one commonly seen in introductory texts, but both methods produce equivalent descriptions of the electron distribution in a molecule.

In order to better understand the bent bond model let's first consider its application to cyclopropane and then move to ethylene.

In cyclopropane it has been found that significant electron density lies off the internuclear axis, rather than along the axis.



(image source)

Further, the $\ce{H-C-H}$ angle in cyclopropane has been measured and found to be 114 degrees.  From this, and using Coulson's Theorem $$\ce{1+\lambda^2cos(114)=0}$$ where $\ce{\lambda^2}$ represents the hybridization index of the bond, the $\ce{C-H}$ bonds in cyclopropane can be deduced to be $\ce{sp^{2.46}}$ hybridized.  Now, using the equation $$\ce{\frac{2}{1+\lambda^_{C-H}^2}+\frac{2}{1+\lambda_{C-C}^2}=1}$$ (which says that summing the "s" character in all bonds at a given carbon must total to 1) we find that $\ce{\lambda_{c-c}^2~=~}$$\mathrm{3.74}$, or the $\ce{C-C}$ bond is $\ce{sp^{3.74}}$ hybridized.  

Pictorially, the bonds look as follows.  They are bent (hence the strain in cyclopropane) and concentrate their electron density off of the internuclear axis as experimentally observed.



We can apply these same concepts to the description of ethylene. Using the known $\ce{H-C-H}$ bond angle of 117 degrees, Coulson's theorem and assuming that we have one p-orbital on each carbon (Hückel's $\sigma - \pi$ formalism), we would conclude that the carbon orbitals involved in the $\ce{C-H}$ bond are $\ce{sp^{2.2}}$ hybridized and the one carbon orbital involved in the $\ce{C-C}$ sigma bond is $\ce{sp^{1.7}}$ hybridized (plus there is the unhybridized p-orbital). This is the "$\ce{sp^2}$" description we see in most textbooks.

Alternately, if we only change our assumption of one pi bond and one sigma bond between the two carbon atoms to two equivalent sigma bonds (Pauling's bent bond formalism), we would find that the carbon orbitals involved in the $\ce{C-H}$ bond are again $\ce{sp^{2.2}}$ hybridized, but the two equivalent $\ce{C-C}$ sigma bonds are $\ce{sp^{4.3}}$ hybridized. We have constructed a two-membered ring cycloalkane analogue!



Hybridization is just a mathematical construct, another model to help us understand and describe molecular bonding.  As shown in the ethylene example above, we can mix carbon's atomic s- and p-orbitals in different ways to describe ethylene. However, the two different ways we mixed these orbitals


one s-orbital plus two p-orbitals plus one unmixed p-orbital (Hückel's  $\sigma - \pi$), or
two s-orbitals plus 2 p-orbitals (Pauling's bent bond)


must lead to equivalent electronic descriptions of ethylene. As noted in the Wikipedia article (first link at the beginning of this answer),


  There is still some debate as to which of the two representations is
  better,[10] although both models are mathematically equivalent. In a
  1996 review, Kenneth B. Wiberg concluded that "although a conclusive
  statement cannot be made on the basis of the currently available
  information, it seems likely that we can continue to consider the σ/π
  and bent-bond descriptions of ethylene to be equivalent.2 Ian
  Fleming goes further in a 2010 textbook, noting that "the overall
  distribution of electrons [...] is exactly the same" in the two
  models.[11]


The bent bond model has the advantage of explaining both cyclopropane (and other strained molecules) as well as olefins.  The strain in cyclopropane and ethylene (e.g. heats of hydrogenation) also make intuitive sense with the bent bond model where the term "bent bond" conjures up an image of strain.

So, bent bonds are a mix of both sigma and pi properties and multiple bent bonds  can be used to describe the bonding between adjacent atoms.