It is not possible to be both a Buddhist and a Christian because Buddhism teaches all things without exception are '*anatta*' ('not-self') & 'the elements' ('*dhatu*') as its core & fundamental principle; where as Christianity includes 'self' in the form of Christ & The Father as a core principle. 

In other words, Buddhism is strictly 'impersonal' where as Christianity is 'personal', where salvation is based on a personal relationship between the 'small-self' & a personal god (Christ & The Father). 

For example, Matthew 16:25 states:

> *Whoever loses his life for **My** sake will find it.*

Luke 23:42 states:

> *Then he said, “Jesus, remember **me** when **You** come into **Your** kingdom!” And Jesus said to him, “Truly I tell **you**, today **you** will be with **Me** in
> Paradise.”…*
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Buddhism explains there are many types of liberation/salvation of mind (MN 43), which include liberation via loving-kindness & liberation via emptiness of self, the later being the foremost & unshakeable. 

> *He keeps pervading above, below & all around, everywhere & in every respect the all-encompassing world with an awareness imbued with good
> will (loving-kindness): abundant, expansive, immeasurable, free from hostility, free
> from ill will... This is called the immeasurable mind-release.*
> 
> *And what is the emptiness mind-release? There is the case where a monk, having gone into the wilderness, to the root of a tree, or
> into an empty dwelling, considers this: 'This is empty of self or of
> anything pertaining to self.'*
> 
> *Now, to the extent that there is immeasurable mind-release, the unprovoked mind-release is declared the foremost. And this
> unprovoked mind-release is empty of passion, empty of aversion,
> empty of delusion.* 

><a href="http://www.accesstoinsight.org/tipitaka/mn/mn.043.than.html">MN 43</a>

Christianity is merely liberation via loving-kindness where as Buddhism is liberation via emptiness of self. 
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Also, Christianity teaches salvation is 'Eternal Life' where as Buddhism teaches salvation is 'dispassion' that occurs via the realisation of impermanence & not-self. In other words, Christianity is a doctrine of 'Eternalism' (the belief a self exists after death), which is a wrong view in Buddhism; where as Buddhism is a doctrine of impermanence. 
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In its most esoteric interpretation (which is rejected by most Christians), Christianity is 'mysticism' where as Buddhism is 'straightforward'. 

Jesus said he taught using parables to hide teachings & the New Testament is replete with the language of metaphor, which results in most Christians being unable to discern between the literal & metaphorical. 

Where as the Buddha taught straightforwardly & plainly, with key terms explicitly defined. 
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Also, Christianity is fatally connected to the Old Testament of Judaism, which allows killing in many circumstances, such as war & heresy. This is why Christianity has had many Inquisitions. 
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In conclusion, Buddhism states it is "impossible" for a person of right view to take refuge in another teacher, thus it is not possible to be both a Buddhist and a Christian.


> *He understands: ‘It is impossible, it cannot happen that a person possessing right view could acknowledge another teacher ― there is no
> such possibility.’ And he understands: ‘It is possible that an
> ordinary person might acknowledge another teacher ― there is such a
> possibility.’*
> 
> *He understands: ‘It is impossible, it cannot happen that two Accomplished Ones, Fully Enlightened Ones, could arise
> contemporaneously in one world-system ― there is no such possibility.’
> And he understands: ‘It is possible that one Accomplished One, a Fully
> Enlightened One, might arise in one world-system ― there is such a
> possibility.*
> 
><a href="http://www.yellowrobe.com/component/content/article/120-majjhima-nikaya/321-bahudhtuka-sutta-the-many-kinds-of-elements.html">MN 115</a>