Can lasers be combined to achieve higher power? For example, if an object's atoms require light with 100 nm to become ionized, can four 400 nm lasers concentrated at one point on the object achieve ionization?  Or will the combined power vary based on different factors?
 A: Yes, very high power laser radiation can cause nonlinear effects, such as multi-photon ionization (https://en.wikipedia.org/wiki/Photoionization#Multi-photon_ionization). Actually, not only four 400 nm lasers, but also just one high-power 400 nm laser can ionize atoms that normally require 100 nm for ionization (https://www.mpi-hd.mpg.de/imprs-qd/fileadmin/user_upload/Internal_School_2013/IMPRS_2013_CMueller.pdf). 
A: 
..., if an object's atoms require light with 100 nm to become ionized, can four 400 nm lasers concentrated at one point on the object achieve ionization?

No they cannot unless we're talking extremely high powers (see Akhmeteli's Answer). This is the essence of the Photoelectric effect: that ionization (or any electronic transition between bound states) requires a threshold shortness of wavelength to make it happen. The intensity of the radiation concerned only affects the rate at which transition events happen. These basic experimental observations can be explained by assuming that the EM field can only transfer energy to electrons in discrete bundles of amount $h\,\nu$, where $h$ is Planck's Quantum of Action (aka Planck's Constant) and $\nu$ the light frequency. When the transition event in question is ionization, the threshold energy needed for a single event, i.e. the binding energy of the electron to the atom concerned,  is called the Work Function. 

Further Question from OP

Since you stated that only high powered lasers can ionize with longer wavelengths than would be required, is there a way of calculating how powerful the laser would need to be? If we still use the 400 nm laser as an example, how powerful would that laser need to be to ionize the atom that requires 100 nm light?

It's the intensity rather than power that does it: generally multiphoton processes are highly concentrated around a focus of a system. You would need to look up the five or six photon cross section of the ion in question at the wavelength in question, and work it out the six-photon event probability and rate from the intensity at the focus. Note that I say five or six photon process: the photon energies don't add linearly for multiphoton processes owing to the weird mechanics of virtual states that mediate these processes. Pulsed lasers with very low duty cycle fraction are used to excite multiphoton processes. For two photon processes, typically hundreds of milliwatts at a focussed through a 0.3NA objective or higher give significant two photon processes in a micron-sized region at the focus when 5 femtosecond pulses are used at 10MHz pulse repetition rate (i.e. duty cycle of the order of $10^{-7}$). For each pulse, this is equivalent to about $10{\rm MW}$ through the focus, corresponding to an intensity of about $10^{19}{\rm W\,m^{-2}}$ at the focus. Four photon processes are orders of magnitude less probable, so we're probably talking about kilowatt / megawatt femtosecond lasers pulsed with a $10^{-7}$ duty cycle. 
A: Be careful not to confuse power with wavelength. The power is just the energy per second carried by the laser beam e.g. a $1$ watt laser carries $1$ joule per second.
The wavelength is related to the energy per photon i.e. the energy per photon is $E = hc/\lambda$.
Ionisation of an atom is (normally) done by a single photon so it requires a certain minimum energy per photon i.e. the wavelength has to be below some upper limit. So if your atom requires light with a $100$ nm wavelength to ionise it no number of $400$ nm lasers is going to achieve the trick. Even though combining four lasers gives four times the power it won't change the wavelength so it won't change the energy per photon.
